Fundamental Functions
What We Will Learn in This Section
In this section, we will study some fundamental functions. These functions serve as the building blocks for other, more complex functions, which are formed by performing various operations (such as addition, multiplication, composition, etc.) on these fundamental functions.
Our focus will be on understanding the domain, range, graph, and other key characteristics of these fundamental functions. Mastery of these basics is essential, as they form the foundation for the study of all functions in mathematics.
Real-Valued Function
All functions in this chapter are real valued functions. A real-valued function is a function where the output values (or range) are all real numbers. In other words, if \( f \) is a function, then for every input \( x \) in the domain of \( f \), the output \( f(x) \) is a real number.
Definition:
A function \( f \) is called a real-valued function if:
where:
- \( A \) is the domain, which may be a subset of real numbers (\( \mathbb{R} \)) or other sets like \( \mathbb{R}^n \).
- \( \mathbb{R} \) is the range, representing all real numbers.
Examples
-
\( f(x) = x^2 \)
- Here, the domain is \( \mathbb{R} \) (all real numbers), and the range is \( [0, \infty) \), which is a subset of \( \mathbb{R} \).
- For every real number \( x \), \( f(x) \) is also a real number.
-
\( g(x) = \sqrt{x} \)
- The domain is \( [0, \infty) \), and the range is \( [0, \infty) \).
- Again, all output values \( g(x) \) are real numbers.
-
\( h(x) = \frac{1}{x} \)
- The domain is \( \mathbb{R} \setminus \{0\} \) (all real numbers except 0), and the range is \( \mathbb{R} \setminus \{0\} \).
- All outputs are real numbers.
Non-Example
A function is not a real-valued function if its output values are not real numbers. For example:
- \( f(x) = \sqrt{-x} \), where \( x > 0 \), is not real-valued because it involves imaginary numbers.
- Complex-valued functions like \( f(x) = e^{ix} \) are not real-valued.
Fundamental Functions
Constant Function
A constant function is a function that assigns the same output value (a constant) to every input in its domain. Regardless of the input, the output remains unchanged.
Definition:
A function \( f(x) \) is called a constant function if it can be written in the form:
where \( c \) is a constant (a fixed real number), and \( x \) belongs to the domain of the function.
Domain and Range
-
Domain: The domain of a constant function is typically \( \mathbb{R} \) (all real numbers), unless specified otherwise.
-
Range: The range of a constant function is the single value \( \{c\} \), which is the constant output of the function.
Example
- Domain: \( \mathbb{R} \) (all real numbers).
- Range: \( \{5\} \) (the output is always 5, regardless of the input).
Example
- Domain: \( \mathbb{R} \) (all real numbers).
- Range: \( \{-3\} \) (the output is always -3).
Graph of a Constant Function
The graph of a constant function is a horizontal line in the Cartesian plane. For example:
- For \( f(x) = 5 \), the graph is a horizontal line at \( y = 5 \).
- For \( g(x) = -3 \), the graph is a horizontal line at \( y = -3 \).
Characteristics of a Constant Function
- Independent of Input: The value of \( f(x) \) does not depend on \( x \).
- Horizontal Line: The graph is always parallel to the \( x \)-axis.
- Range is a Single Value: The output value is always constant, forming a one-element set.
Identity Function
An identity function is a function where the output is always equal to the input. It is one of the simplest types of functions and serves as a fundamental example in mathematics.
Definition:
A function \( f(x) \) is called an identity function if:
where \( x \) belongs to the domain of the function.
In this case, for every input \( x \), the output \( f(x) \) is the same as \( x \).
Domain and Range:
- Domain: The domain of the identity function is typically \( \mathbb{R} \) (all real numbers), unless specified otherwise.
- Range: The range is also \( \mathbb{R} \), as the output \( f(x) = x \) can take any real value.
Graph of the Identity Function:
- The graph of \( f(x) = x \) is a straight line passing through the origin \( (0, 0) \), with a slope of 1.
- This line lies at a \( 45^\circ \) angle with respect to the \( x \)-axis and bisects the first and third quadrants.
Example
-
Point Evaluation:
- For \( x = -2 \), \( f(-2) = -2 \).
- For \( x = 0 \), \( f(0) = 0 \).
- For \( x = 3 \), \( f(3) = 3 \).
-
Graph Description:
- Passes through points like \( (-2, -2) \), \( (0, 0) \), \( (3, 3) \), and so on.
- Every point on the graph satisfies \( y = x \).
Characteristics of the Identity Function:
- Input Equals Output: The value of \( f(x) \) is always the same as \( x \).
- Domain and Range Are Equal: Both are \( \mathbb{R} \).
- Straight Line: The graph is a straight line with a slope of 1.
- Symmetry: The graph is symmetric about the line \( y = x \), as it lies directly on this line.
Modulus Function
The modulus function, also called the absolute value function, is a function that gives the non-negative value (magnitude) of its input, regardless of whether the input is positive or negative. It is defined as follows:
Definition:
- When \( x \) is non-negative (\( x \geq 0 \)), the function returns \( x \).
- When \( x \) is negative (\( x < 0 \)), the function returns \( -x \), which makes it positive.
Domain and Range:
-
Domain: The domain of \( f(x) = |x| \) is \( \mathbb{R} \) (all real numbers) since the function is defined for all real values of \( x \).
-
Range: The range of \( f(x) = |x| \) is \( [0, \infty) \), as the output is always non-negative.
Graph of the Modulus Function:
- The graph of \( f(x) = |x| \) is V-shaped.
- For \( x \geq 0 \), \( f(x) = x \), so the graph is a straight line passing through the origin with a slope of 1.
- For \( x < 0 \), \( f(x) = -x \), so the graph is a straight line passing through the origin with a slope of -1.
Key Points on the Graph:
- The graph passes through points like \( (-2, 2) \), \( (0, 0) \), \( (3, 3) \), etc.
- The point \( (0, 0) \) is the vertex of the graph, where the two linear pieces meet.
Characteristics of the Modulus Function:
- Piecewise Definition: The function behaves differently for \( x \geq 0 \) and \( x < 0 \).
- Non-Negativity: The output is always \( \geq 0 \).
- Symmetry: The function is symmetric about the \( y \)-axis (even function).
- Continuous: The function has no breaks or gaps in its graph.
Linear Function
A linear function is a function whose graph is a straight line in the Cartesian plane. It is one of the simplest types of functions and is represented by the equation:
where:
- \( m \) is the slope of the line, representing its steepness.
- \( c \) is the y-intercept, the point where the line crosses the \( y \)-axis.
Domain and Range:
-
Domain: The domain of a linear function is \( \mathbb{R} \) (all real numbers), as the function is defined for every real value of \( x \).
-
Range: If \(m\ne0\), then The range of a linear function is also \( \mathbb{R} \), as the output \( f(x) \) can take any real value depending on \( x \). If \(m=0\), \(f\) is a constant function and thus the range is \(\{c\}\).
Example
-
Function: Let \( f(x) = 2x + 3 \).
- Slope: \( m = 2 \), which indicates the line rises 2 units for every 1 unit increase in \( x \).
- y-Intercept: \( c = 3 \), meaning the line crosses the \( y \)-axis at \( (0, 3) \).
-
Evaluation:
- At \( x = 0 \): \( f(0) = 2(0) + 3 = 3 \).
- At \( x = 1 \): \( f(1) = 2(1) + 3 = 5 \).
- At \( x = -2 \): \( f(-2) = 2(-2) + 3 = -1 \).
Graph of a Linear Function:
- The graph of \( f(x) = mx + c \) is a straight line.
- The slope \( m \) determines the tilt of the line:
- \( m > 0 \): Line rises as \( x \) increases.
- \( m < 0 \): Line falls as \( x \) increases.
- \( m = 0 \): Line is horizontal (constant function).
- The \( y \)-intercept \( c \) determines where the line crosses the \( y \)-axis.
Square Function
The square function is a fundamental type of polynomial function, defined as:
where the output is the square of the input.
Domain and Range
-
Domain:
- The domain of \( f(x) = x^2 \) is \( \mathbb{R} \) (all real numbers), as squaring any real number produces a valid result.
-
Range:
- The range of \( f(x) = x^2 \) is \( [0, \infty) \), since \( x^2 \) is always non-negative, regardless of whether \( x \) is positive, negative, or zero.
Graph of the Square Function
Using the following key points on the graph, we can easily draw \(f(x)=x^2\):
These points \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), \((2, 4)\) help form the graph.
- The graph of \( f(x) = x^2 \) is a parabola that opens upwards.
- The vertex (lowest point) of the parabola is at \( (0, 0) \).
- The parabola is symmetric about the \( y \)-axis, as \( f(-x) = (-x)^2 = x^2 \).
Cubic Function
The cubic function is a polynomial function of degree 3, defined as:
where the output is the cube of the input.
Domain and Range
-
Domain:
- The domain of \( f(x) = x^3 \) is \( \mathbb{R} \) (all real numbers), as cubing any real number produces a valid result.
-
Range:
- The range of \( f(x) = x^3 \) is also \( \mathbb{R} \) (all real numbers), since \( x^3 \) can take any real value, both positive and negative.
Graph of the Cubic Function
These points \((-2, -8)\), \((-1, -1)\), \((0, 0)\), \((1, 1)\), \((2, 8)\) can be used to sketch the graph.
- The graph of \( f(x) = x^3 \) is an S-shaped curve.
- It passes through the origin \( (0, 0) \) and is symmetric about the origin.
- For \( x > 0 \), the graph increases steeply as \( x \) grows.
- For \( x < 0 \), the graph decreases steeply as \( x \) becomes more negative.
Monomial Function
Functions of the Form \( f(x) = x^n \), \( n \in \mathbb{N} \)
In this section, we study the behavior of functions where the variable \( x \) is raised to a natural-number power \( n \). Such functions take the general form \( f(x) = x^n \). Depending on whether \( n \) is even or odd, the properties of the function and its graph exhibit significant differences.
Case I: \( n \) is Even
When \( n \) is an even natural number, such as \( n = 2, 4, 6, \dots \), the function \( f(x) = x^n \) exhibits symmetry about the \( y \)-axis. This is because \( f(-x) = (-x)^n = x^n \), making the function even.
The graph of \( f(x) = x^n \) for even \( n \) always lies above or on the \( x \)-axis. For \( n = 2 \), the graph is a parabola opening upwards. As \( n \) increases, the shape of the graph near \( x = 0 \) becomes flatter, while it steepens significantly as \( |x| \) moves away from 1. This steepening reflects the dominance of higher powers of \( x \) for larger values of \( |x| \).
A notable feature of these graphs is their intersection at three key points: \( (0, 0) \), \( (1, 1) \), and \( (-1, 1) \). For \( -1 < x < 1 \), higher powers \( x^n \) are smaller than lower powers \( x^m \), where \( m < n \). That is, for \( -1 < x < 1 \),
On the other hand, for \( |x| > 1 \), higher powers \( x^n \) exceed lower powers \( x^m \), indicating that the graph becomes steeper as \( n \) increases.
The domain of \( f(x) = x^n \) is the set of all real numbers, \( \mathbb{R} \). However, since \( x^n \geq 0 \) for even \( n \), the range is \( [0, \infty) \).
Case II: \( n \) is Odd
When \( n \) is an odd natural number, such as \( n = 1, 3, 5, \dots \), the function \( f(x) = x^n \) exhibits symmetry about the origin.
For \( n = 1 \), the graph of \( f(x) = x \) is a straight line passing through the origin. As \( n \) increases, the graph retains the general "S" shape but becomes flatter near \( x = 0 \) and steeper as \( |x| \) moves away from 1. The graphs of all odd powers intersect at three common points: \( (0, 0) \), \( (1, 1) \), and \( (-1, -1) \).
The domain of \( f(x) = x^n \) remains \( \mathbb{R} \), and, unlike the even case, the range also spans \( \mathbb{R} \), as the function takes both positive and negative values.
Square Root Function
The square root function, \( f(x) = \sqrt{x} \), is defined only for non-negative values of \( x \), as the square root of a negative number is not real. The function has several important properties, which we detail below:
Domain and Range
-
Domain: The square root function is defined for all \( x \geq 0 \). Hence, the domain is:
\[ \text{Domain} = [0, \infty). \] -
Range: The output of \( \sqrt{x} \) is also non-negative, since a square root cannot yield a negative value. Therefore, the range is:
\[ \text{Range} = [0, \infty). \]
Graphical Behavior
The graph of \( f(x) = \sqrt{x} \) starts at the origin \( (0, 0) \) and increases monotonically. The rate of increase, however, slows down as \( x \) becomes larger. This behavior can be observed in the following two distinct regions:
-
Near the Origin (\( x \to 0^+ \)): As \( x \) approaches \( 0 \) from the right, \( f(x) = \sqrt{x} \) also approaches \( 0 \). However, the graph rises steeply for small values of \( x \), exhibiting a nearly vertical slope. This steep rise reflects the rapid increase of \( \sqrt{x} \) for values of \( x \) close to \( 0 \).
-
As \( x \to \infty \):
As \( x \) grows larger, \( \sqrt{x} \) increases, but at a much slower rate. For instance:
\[ f(100) = \sqrt{100} = 10, \quad f(10,000) = \sqrt{10,000} = 100. \]
Cube Root Function
The cube root function, \( f(x) = \sqrt[3]{x} \), is defined for all real numbers \( x \), since the cube root of a number always exists, whether the number is positive, negative, or zero. This function has distinct properties, which are discussed below.
Domain and Range
-
Domain: The cube root function is defined for all \( x \in \mathbb{R} \). Thus, the domain is:
\[ \text{Domain} = (-\infty, \infty). \] -
Range: Since \( \sqrt[3]{x} \) can take any real value depending on \( x \), the range is:
\[ \text{Range} = (-\infty, \infty). \]
Graphical Behavior
The graph of \( f(x) = \sqrt[3]{x} \) passes through the origin and is symmetric about the origin, as the function satisfies \( f(-x) = -f(x) \). The behavior of the graph differs in three regions:
-
Near the Origin (\( x \to 0 \)):
As \( x \to 0 \), \( f(x) = \sqrt[3]{x} \) also approaches \( 0 \). The graph has a steep slope near the origin, as small values of \( x \) result in relatively large changes in \( f(x) \).
-
For Large Positive \( x \) (\( x \to \infty \)):
As \( x \to \infty \), \( f(x) = \sqrt[3]{x} \) also approaches \( \infty \). However, the growth is slow compared to linear or higher-degree polynomial functions. For example:
\[ f(1000) = \sqrt[3]{1000} = 10, \quad f(1,000,000) = \sqrt[3]{1,000,000} = 100. \] -
For Large Negative \( x \) (\( x \to -\infty \)):
As \( x \to -\infty \), \( f(x) = \sqrt[3]{x} \) approaches \( -\infty \). The function exhibits similar slow growth in magnitude for large negative \( x \), but in the negative direction. For example:
\[ f(-1000) = \sqrt[3]{-1000} = -10, \quad f(-1,000,000) = \sqrt[3]{-1,000,000} = -100. \]
Graphical Insight
The graph of \( f(x) = \sqrt[3]{x} \) passes through the origin and has a smooth, continuous curve. For positive values of \( x \), the function rises slowly as \( x \to \infty \). For negative values of \( x \), the graph falls into the third quadrant, approaching \( -\infty \) as \( x \to -\infty \). The steepness of the graph near the origin reflects the rapid change of \( \sqrt[3]{x} \) for small values of \( x \), while the flatness for large \( |x| \) demonstrates the slow growth characteristic of cube roots.
Reciprocal Function
The reciprocal function \( f(x) = \frac{1}{x} \) is a foundational example of a function involving division. It is defined for all real numbers \( x \) except \( x = 0 \), as division by zero is undefined.
Domain and Range
-
Domain: The function \( f(x) = \frac{1}{x} \) requires that \( x \neq 0 \). Therefore, the set of all permissible values of \( x \) is:
\[ \text{Domain} = (-\infty, 0) \cup (0, \infty). \] -
Range: The values of \( f(x) \) are all real numbers except \( 0 \), since dividing \( 1 \) by any nonzero number never results in zero. Hence, the range is:
\[ \text{Range} = (-\infty, 0) \cup (0, \infty). \]
Graphical Behavior
The graph of \( f(x) = \frac{1}{x} \) has two distinct branches, depending on whether \( x > 0 \) or \( x < 0 \). Let us analyze these cases:
-
For Positive \( x \) (\( x > 0 \)):
- When \( x > 0 \), the value of \( f(x) = \frac{1}{x} \) is also positive.
-
As \( x \) becomes very small but remains positive (close to \( 0 \)), the value of \( f(x) \) becomes very large. For instance:
\[ f(0.1) = \frac{1}{0.1} = 10, \quad f(0.01) = \frac{1}{0.01} = 100. \]Thus, the graph rises steeply as \( x \to 0^+ \).
-
As \( x \) becomes very large, \( f(x) \) becomes very small. For example:
\[ f(10) = \frac{1}{10} = 0.1, \quad f(100) = \frac{1}{100} = 0.01. \]This indicates that the graph flattens as \( x \to \infty \).
-
For Negative \( x \) (\( x < 0 \)):
- When \( x < 0 \), the value of \( f(x) = \frac{1}{x} \) is negative.
-
As \( x \) becomes very small in magnitude but remains negative (close to \( 0 \)), the value of \( f(x) \) becomes very large in magnitude but remains negative. For instance:
\[ f(-0.1) = \frac{1}{-0.1} = -10, \quad f(-0.01) = \frac{1}{-0.01} = -100. \]Thus, the graph falls steeply as \( x \to 0^- \).
-
As \( x \) becomes very large in magnitude but remains negative, \( f(x) \) becomes very small in magnitude but remains negative. For example:
\[ f(-10) = \frac{1}{-10} = -0.1, \quad f(-100) = \frac{1}{-100} = -0.01. \]This shows that the graph flattens as \( x \to -\infty \).
Key Observations
-
The graph has two distinct parts: one in the first quadrant (\( x > 0, f(x) > 0 \)) and one in the third quadrant (\( x < 0, f(x) < 0 \)).
-
The graph approaches but never touches the \( x \)-axis or the \( y \)-axis:
- The \( y \)-axis corresponds to \( x = 0 \), where the function is not defined.
- The \( x \)-axis corresponds to \( f(x) = 0 \), which is never attained by the function.
This behavior—where the function gets arbitrarily close to a line without actually touching it—will later lead us to the concept of asymptotes, which describe such tendencies rigorously.
The reciprocal function’s unique structure highlights the concept of inverse proportionality: as \( x \) increases, \( f(x) \) decreases, and vice versa.
Reciprocal Squared Function
The reciprocal squared function \( f(x) = \frac{1}{x^2} \) is defined for all \( x \neq 0 \) and exhibits distinct growth and decay patterns depending on the value of \( x \). Let us analyze its properties in detail.
Domain and Range
-
Domain:
The function \( f(x) = \frac{1}{x^2} \) is undefined at \( x = 0 \), as division by zero is not possible. Thus, the domain is:
\[ \text{Domain} = (-\infty, 0) \cup (0, \infty). \] -
Range:
Since \( x^2 > 0 \) for all \( x \neq 0 \), \( f(x) = \frac{1}{x^2} \) is always positive. The function approaches \( 0^+ \) for large values of \( |x| \), and \( f(x) \to \infty \) as \( |x| \to 0 \). Thus, the range is:
\[ \text{Range} = (0, \infty). \]
Graphical Behavior
-
For Positive \( x \) (\( x > 0 \)):
- When \( x > 0 \), \( f(x) = \frac{1}{x^2} > 0 \).
- As \( x \to 0^+ \), \( f(x) \to \infty \), indicating that the graph rises steeply near the \( y \)-axis.
- As \( x \to \infty \), \( f(x) \to 0^+ \), causing the graph to flatten and approach the \( x \)-axis.
-
For Negative \( x \) (\( x < 0 \)):
- When \( x < 0 \), \( f(x) = \frac{1}{x^2} > 0 \), since \( (-x)^2 = x^2 \).
- As \( x \to 0^- \), \( f(x) \to \infty \), showing a steep rise near the \( y \)-axis.
- As \( x \to -\infty \), \( f(x) \to 0^+ \), causing the graph to flatten and approach the \( x \)-axis.
-
Symmetry:
The function \( f(x) = \frac{1}{x^2} \) is symmetric about the \( y \)-axis because \( f(-x) = f(x) \). The graph lies entirely in the first and second quadrants.
End Behavior
-
As \( x \to \infty \): The value of \( f(x) \) approaches \( 0^+ \). For example:
\[ f(10) = \frac{1}{10^2} = 0.01, \quad f(100) = \frac{1}{100^2} = 0.0001. \] -
As \( x \to -\infty \): The value of \( f(x) \) also approaches \( 0^+ \). For example:
\[ f(-10) = \frac{1}{(-10)^2} = 0.01, \quad f(-100) = \frac{1}{(-100)^2} = 0.0001. \] -
As \( x \to 0^+ \): The value of \( f(x) \) increases without bound, \( f(x) \to \infty \). For example:
\[ f(0.1) = \frac{1}{(0.1)^2} = 100, \quad f(0.01) = \frac{1}{(0.01)^2} = 10,000. \] -
As \( x \to 0^- \): The value of \( f(x) \) also increases without bound, \( f(x) \to \infty \). For example:
\[ f(-0.1) = \frac{1}{(-0.1)^2} = 100, \quad f(-0.01) = \frac{1}{(-0.01)^2} = 10,000. \]
The reciprocal squared function \( f(x) = \frac{1}{x^2} \) has two branches, one in the first quadrant and the other in the second quadrant. Both branches rise steeply as \( x \to 0 \) and flatten as \( |x| \to \infty \). Its distinct behavior near \( 0 \) and at infinity highlights the interplay between rapid growth and slow decay.
Graph of General Monomials
The monomials function \( f(x) = ax^n \) exhibits distinct behavior depending on whether \( n \) (the exponent) is even or odd and whether \( a \) (the coefficient) is positive or negative. Let us analyze the end behavior for both cases.
Case 1: \( n \) Even
When \( n \) is even (\( n = 2, 4, 6, \dots \)), the graph is symmetric about the \( y \)-axis because \( (-x)^n = x^n \). The sign of \( a \) determines the direction in which the graph opens.
-
When \( a > 0 \):
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).
- The graph rises on both ends, forming a "U" shape.
-
When \( a < 0 \):
- As \( x \to \infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- The graph falls on both ends, forming an upside-down "U" shape.
Case 2: \( n \) Odd
When \( n \) is odd (\( n = 1, 3, 5, \dots \)), the graph is not symmetric about the \( y \)-axis but exhibits origin symmetry because \( (-x)^n = -x^n \). The sign of \( a \) determines the direction of the graph at each end.
-
When \( a > 0 \):
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- The graph falls on the left and rises on the right, resembling an "S" shape.
-
When \( a < 0 \):
- As \( x \to \infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).
- The graph rises on the left and falls on the right, forming a mirrored "S" shape.
Summary of End Behavior
| Exponent \( n \) | Coefficient \( a > 0 \) | Coefficient \( a < 0 \) |
|---|---|---|
| \( n \) Even | \( f(x) \to \infty \) as \( x \to \pm\infty \) | \( f(x) \to -\infty \) as \( x \to \pm\infty \) |
| \( n \) Odd | \( f(x) \to \infty \) as \( x \to \infty \), \( f(x) \to -\infty \) as \( x \to -\infty \) | \( f(x) \to -\infty \) as \( x \to \infty \), \( f(x) \to \infty \) as \( x \to -\infty \) |
Graphical Behavior
-
For \( n \) Even:
- The graph is symmetric about the \( y \)-axis.
- The shape resembles a parabola for \( n = 2 \), but for higher \( n \), the graph flattens near \( x = 0 \) and steepens near \( |x| \to \infty \).
-
For \( n \) Odd:
- The graph exhibits origin symmetry.
- For small \( |x| \), the graph flattens, while for large \( |x| \), it steepens.
Polynomial Functions
A polynomial function \( p(x) = a_0x^n + a_1x^{n-1} + \cdots + a_n \) of degree \( n \) is determined by its terms, coefficients, and powers of \( x \). The most significant term is the leading term, \( a_0x^n \), which dominates the behavior of the polynomial for large values of \( |x| \). The leading coefficient \( a_0 \) and the degree \( n \) primarily determine the polynomial's end behavior.
We begin by expressing \( p(x) \) in a form that emphasizes its dominant term:
As \( |x| \to \infty \), the terms \( \frac{a_1}{a_0x}, \frac{a_2}{a_0x^2}, \dots \) approach \( 0 \). This implies that for large \( |x| \), the polynomial approximates:
Thus, the leading term \( a_0x^n \) determines the behavior of \( p(x) \) at the extremes of the domain.
The behavior near \( x = 0 \) is more complex, as the lower-order terms \( a_1x^{n-1}, a_2x^{n-2}, \dots \) play a significant role. The interaction of these terms with the higher-order terms often results in oscillations, or "up-and-down" behavior, near the origin.
End Behavior Based on Degree and Leading Coefficient
For \( n \) even, the powers of \( x \) are symmetric about the \( y \)-axis, as \( (-x)^n = x^n \). If \( a_0 > 0 \), the polynomial increases to \( \infty \) as \( x \to \infty \) and as \( x \to -\infty \). This results in a graph that rises on both ends, forming a U-shaped curve. Conversely, if \( a_0 < 0 \), the polynomial decreases to \( -\infty \) as \( x \to \infty \) and as \( x \to -\infty \), resulting in an inverted U-shaped graph.
For \( n \) odd, the powers of \( x \) satisfy \( (-x)^n = -x^n \), leading to symmetry about the origin. If \( a_0 > 0 \), the polynomial increases to \( \infty \) as \( x \to \infty \) and decreases to \( -\infty \) as \( x \to -\infty \). This creates a graph that falls on the left and rises on the right. On the other hand, if \( a_0 < 0 \), the polynomial decreases to \( -\infty \) as \( x \to \infty \) and increases to \( \infty \) as \( x \to -\infty \), resulting in a graph that rises on the left and falls on the right.
These patterns of end behavior are summarized as follows:
-
For \( n \) even:
- \( a_0 > 0 \): \( p(x) \to \infty \) as \( x \to \pm\infty \).
- \( a_0 < 0 \): \( p(x) \to -\infty \) as \( x \to \pm\infty \).
-
For \( n \) odd:
- \( a_0 > 0 \): \( p(x) \to \infty \) as \( x \to \infty \) and \( p(x) \to -\infty \) as \( x \to -\infty \).
- \( a_0 < 0 \): \( p(x) \to -\infty \) as \( x \to \infty \) and \( p(x) \to \infty \) as \( x \to -\infty \).
Range of the Polynomial Function
The range of \( p(x) \) depends on whether \( n \) is even or odd and the sign of \( a_0 \).
-
For \( n \) Even:
- If \( a_0 > 0 \), the graph rises on both ends, and the range is \( [M, \infty) \), where \( M \) is the minimum value of \( p(x) \), typically at the vertex of the curve.
- If \( a_0 < 0 \), the graph falls on both ends, and the range is \( (-\infty, M] \), where \( M \) is the maximum value of \( p(x) \), typically at the vertex.
-
For \( n \) Odd:
-
The graph extends indefinitely in both the positive and negative directions. Hence, the range is:
\[ \text{Range} = \mathbb{R}. \]
-
Graphical Insights
-
For Large \( |x| \):
The graph of \( p(x) \) closely resembles the graph of \( a_0x^n \).
-
For Small \( |x| \):
The interaction of the terms with coefficients \( a_1, a_2, \dots, a_n \) causes local fluctuations in the graph, often leading to turning points and changes in direction.
-
Turning Points:
A polynomial of degree \( n \) can have up to \( n-1 \) turning points, which are influenced by the interplay of all the coefficients.
This structure provides a comprehensive understanding of how \( p(x) \) behaves for both large and small values of \( x \), based on its degree \( n \) and leading coefficient \( a_0 \), along with the range determined by \( n \) being even or odd.
Graph of Polynomial Functions in Factorized Form
The polynomial function \( p(x) = (x-\alpha_1)^{n_1}(x-\alpha_2)^{n_2}\cdots(x-\alpha_k)^{n_k} \) is characterized by its roots \( \alpha_1, \alpha_2, \dots, \alpha_k \) and their corresponding multiplicities \( n_1, n_2, \dots, n_k \). The leading term, obtained from the highest power of \( x \) in the expanded polynomial, determines the degree \( n \) of \( p(x) \) and its end behavior. The interaction of the graph with the \( x \)-axis at each root depends on the power of the corresponding factor.
To explore this, consider the examples \( p_1(x) = (x-1)(x-2)(x-3) \) and \( p_2(x) = (x-1)^2(x-2)(x-3) \).
Leading Term and End Behavior
For \( p_1(x) \), the degree is \( n = 3 \), as the leading term is \( x^3 \). This means the end behavior of the graph is determined by the function \( y = x^3 \). As \( x \to \infty \), \( p_1(x) \to \infty \), and as \( x \to -\infty \), \( p_1(x) \to -\infty \). For \( p_2(x) \), the degree is \( n = 4 \), as the leading term is \( x^4 \). In this case, the end behavior of the graph is like \( y = x^4 \), so as \( x \to \pm\infty \), \( p_2(x) \to \infty \). Thus, for large \( |x| \), \( p_1(x) \) approximates \( x^3 \), while \( p_2(x) \) approximates \( x^4 \).
Behavior at Roots
Both \( p_1(x) \) and \( p_2(x) \) share the same roots, \( x = 1, 2, 3 \). At these roots, the graph interacts with the \( x \)-axis depending on the multiplicity of the factor. For \( p_1(x) \), each root has multiplicity 1, which is odd. Therefore, the graph crosses the \( x \)-axis at \( x = 1, x = 2, \) and \( x = 3 \). In contrast, for \( p_2(x) \), the root at \( x = 1 \) has multiplicity 2, which is even. This means the graph touches the \( x \)-axis at \( x = 1 \) without crossing it, while it crosses the \( x \)-axis at \( x = 2 \) and \( x = 3 \), where the multiplicities are odd.
Graphical Construction
The graph of \( p_1(x) = (x-1)(x-2)(x-3) \) alternates above and below the \( x \)-axis as it moves from \( x \to -\infty \) to \( x \to \infty \), consistent with the odd degree \( n = 3 \). It crosses the \( x \)-axis at \( x = 1, 2, 3 \). For \( p_2(x) = (x-1)^2(x-2)(x-3) \), the graph rises on both ends due to the even degree \( n = 4 \). It touches the \( x \)-axis at \( x = 1 \) and crosses the axis at \( x = 2 \) and \( x = 3 \).
Example
The polynomial \( p(x) = (x-1)^2(x+1)^3(x+2)(x-3)^2(x-4) \) is in fully factorized form. Its behavior is determined by the degree, the leading term, and the roots along with their multiplicities.
Leading Term and End Behavior
The degree of the polynomial is obtained by summing the powers of all factors:
The leading term of \( p(x) \) is the product of the leading terms of each factor. Since the leading term of each factor is \( x^{n_i} \), the leading term of \( p(x) \) is:
Thus, the end behavior of the graph follows that of \( y = x^9 \). As \( x \to \infty \), \( p(x) \to \infty \), and as \( x \to -\infty \), \( p(x) \to -\infty \).
Roots and Behavior at Each Root
The roots of \( p(x) \) are \( x = 1, -1, -2, 3, \) and \( 4 \), with their respective multiplicities \( 2, 3, 1, 2, \) and \( 1 \). The multiplicities determine how the graph interacts with the \( x \)-axis at each root:
- At \( x = 1 \), the factor \( (x-1)^2 \) has even multiplicity, so the graph touches the \( x \)-axis but does not cross it.
- At \( x = -1 \), the factor \( (x+1)^3 \) has odd multiplicity, so the graph crosses the \( x \)-axis.
- At \( x = -2 \), the factor \( (x+2) \) has odd multiplicity, so the graph crosses the \( x \)-axis.
- At \( x = 3 \), the factor \( (x-3)^2 \) has even multiplicity, so the graph touches the \( x \)-axis but does not cross it.
- At \( x = 4 \), the factor \( (x-4) \) has odd multiplicity, so the graph crosses the \( x \)-axis.
Graphical Construction
The graph of \( p(x) \) alternates above and below the \( x \)-axis as it moves from \( x \to -\infty \) to \( x \to \infty \), consistent with the odd degree \( n = 9 \). Near each root:
- The graph touches the \( x \)-axis at \( x = 1 \) and \( x = 3 \), due to their even multiplicities.
- The graph crosses the \( x \)-axis at \( x = -1, -2, \) and \( 4 \), due to their odd multiplicities.
For large \( |x| \), the graph closely resembles the behavior of \( y = x^9 \), rising steeply to \( \infty \) as \( x \to \infty \) and falling steeply to \( -\infty \) as \( x \to -\infty \).
This example illustrates how the degree, the leading term, and the multiplicities of the factors determine the global and local behavior of a polynomial's graph.
Peculiarities of Polynomial Graphs: Behavior at Roots Based on Factor Powers
When analyzing the graph of a polynomial \( p(x) = (x-\alpha_1)^{n_1}(x-\alpha_2)^{n_2}\cdots(x-\alpha_k)^{n_k} \), another intriguing aspect arises concerning the slope of the graph at the roots, which depends on the power of the corresponding factor \( (x-\alpha_i)^{n_i} \). While the formal proof of this behavior requires knowledge of differentiation, we describe the key observations here.
Behavior at Roots Based on the Power of the Factor
-
When the Power \( n_i = 1 \):
If the corresponding factor \( (x-\alpha_i) \) has a power of 1, the graph crosses the \( x \)-axis at the root \( x = \alpha_i \) with a nonzero slope. This means the graph intersects the \( x \)-axis at an angle. The crossing is linear in nature, reflecting the odd multiplicity of the factor.
-
When the Power \( n_i = 2, 3, 4, \dots \):
For higher powers of \( (x-\alpha_i) \), the behavior changes:
- For \( n_i = 2 \): The graph touches the \( x \)-axis at \( x = \alpha_i \) and becomes flat at that point. This flatness corresponds to the slope of the graph being zero at \( x = \alpha_i \).
- For \( n_i = 3 \): The graph crosses the \( x \)-axis at \( x = \alpha_i \), but the crossing is more gradual compared to \( n_i = 1 \). The slope at \( x = \alpha_i \) approaches zero momentarily, resulting in a slightly flattened crossing.
- For \( n_i = 4, 5, \dots \): As the power increases, the graph becomes progressively flatter near \( x = \alpha_i \). For even \( n_i \), the graph only touches the \( x \)-axis, while for odd \( n_i \), it crosses the \( x \)-axis but with increasing flatness as \( n_i \) increases.
The flatness of the graph at roots where \( n_i \geq 2 \) can be explained through differentiation, which we will explore in detail later. Differentiation allows us to calculate the slope of the polynomial at any point. For a root \( x = \alpha_i \) with a factor \( (x-\alpha_i)^{n_i} \), the derivative of \( p(x) \) reveals that the slope at \( x = \alpha_i \) is zero when \( n_i \geq 2 \). This explains why the graph appears flat at these points.
Exponential Function
An exponential function is defined as \( f(x) = a^x \), where \( a > 0 \) and \( a \neq 1 \). The base \( a \) determines the behavior of the function. For \( a > 1 \), \( f(x) = a^x \) represents exponential growth, while for \( 0 < a < 1 \), \( f(x) = a^x \) represents exponential decay. These functions are distinguished by their rapid growth or decay, their unique response to changes in \( x \), and their critical role in modeling natural phenomena like population growth, radioactive decay, and compound interest.
The structure of \( f(x) = a^x \) reflects several fundamental properties:
- It is always defined for any real \( x \), as raising a positive base \( a \) to any real power is valid.
- It is always strictly positive, never reaching zero or negative values.
- The function passes through \( (0, 1) \) because \( a^0 = 1 \) for any \( a > 0 \), making this a universal feature of exponential functions.
Let us analyze this behavior in detail using examples, \( f_1(x) = 2^x \) and \( f_2(x) = \left(\frac{1}{2}\right)^x \), which illustrate the cases \( a > 1 \) and \( 0 < a < 1 \), respectively.
Example 1: \( f_1(x) = 2^x \)
For \( f_1(x) = 2^x \), the base \( a = 2 > 1 \), so the function exhibits exponential growth. Evaluating \( f_1(x) \) at specific points helps us understand its behavior:
- At \( x = 0 \), \( f_1(0) = 2^0 = 1 \). This confirms that the graph intersects the \( y \)-axis at \( (0, 1) \).
- At \( x = 1 \), \( f_1(1) = 2^1 = 2 \), and at \( x = 2 \), \( f_1(2) = 2^2 = 4 \), showing that the function grows rapidly as \( x \) increases.
- For negative \( x \), \( f_1(-1) = 2^{-1} = \frac{1}{2} \) and \( f_1(-2) = 2^{-2} = \frac{1}{4} \), indicating that the function approaches \( 0 \) as \( x \to -\infty \).
The graph of \( f_1(x) \) rises steeply as \( x \to \infty \) and flattens near the \( x \)-axis as \( x \to -\infty \). This behavior reflects the dominance of the base \( a = 2 \), where each unit increase in \( x \) causes the value of \( f_1(x) \) to double.
Example 2: \( f_2(x) = \left(\frac{1}{2}\right)^x \)
For \( f_2(x) = \left(\frac{1}{2}\right)^x \), the base \( a = \frac{1}{2} < 1 \), so the function exhibits exponential decay. Evaluating \( f_2(x) \) at specific points provides insight into its behavior:
- At \( x = 0 \), \( f_2(0) = \left(\frac{1}{2}\right)^0 = 1 \). Like all exponential functions, the graph intersects the \( y \)-axis at \( (0, 1) \).
- At \( x = 1 \), \( f_2(1) = \frac{1}{2} \), and at \( x = 2 \), \( f_2(2) = \frac{1}{4} \), demonstrating a rapid decay as \( x \) increases.
- For negative \( x \), \( f_2(-1) = \left(\frac{1}{2}\right)^{-1} = 2 \), and \( f_2(-2) = \left(\frac{1}{2}\right)^{-2} = 4 \), indicating that the function grows exponentially as \( x \to -\infty \).
The graph of \( f_2(x) \) falls steeply as \( x \to \infty \) and rises sharply as \( x \to -\infty \). This behavior reflects the inverse relationship between \( x \) and the base \( a = \frac{1}{2} \), where each unit increase in \( x \) causes the value of \( f_2(x) \) to halve.
Key Observations and Characteristics
From these examples, we observe several key features of exponential functions:
- The domain of \( f(x) = a^x \) is all real numbers, \( (-\infty, \infty) \), since the operation \( a^x \) is defined for any real \( x \).
- The range is strictly positive, \( (0, \infty) \), because \( a^x > 0 \) for any \( a > 0 \). Exponential functions never touch or cross the \( x \)-axis, which serves as a horizontal asymptote.
- The point \( (0, 1) \) is a universal feature of exponential functions because \( a^0 = 1 \) for any \( a > 0 \).
- For \( a > 1 \), the function grows exponentially as \( x \to \infty \) and approaches \( 0^+ \) as \( x \to -\infty \). Conversely, for \( 0 < a < 1 \), the function decays exponentially as \( x \to \infty \) and grows exponentially as \( x \to -\infty \).
End Behavior
The end behavior of \( f(x) = a^x \) depends on the value of \( a \): - For \( a > 1 \), as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to 0 \). - For \( 0 < a < 1 \), as \( x \to \infty \), \( f(x) \to 0 \), and as \( x \to -\infty \), \( f(x) \to \infty \).
Logarithmic Functions
A logarithmic function is defined as \( f(x) = \log_a x \), where \( a > 0 \) and \( a \neq 1 \). It represents the inverse of the exponential function \( a^y = x \). The value \( \log_a x \) answers the question: "To what power must \( a \) be raised to produce \( x \)?" This inverse relationship determines the behavior of logarithmic functions, which depends on whether \( a > 1 \) or \( 0 < a < 1 \).
Let us analyze their characteristics using specific examples, \( f_1(x) = \log_2 x \) (base greater than 1) and \( f_2(x) = \log_{1/2} x \) (base between 0 and 1).
Example 1: \( f_1(x) = \log_2 x \)
The function \( f_1(x) = \log_2 x \) is the inverse of \( 2^x \), so \( y = \log_2 x \iff 2^y = x \). To understand its behavior, consider:
- At \( x = 1 \), \( \log_2 1 = 0 \) because \( 2^0 = 1 \). Thus, the graph passes through \( (1, 0) \).
- At \( x = 2 \), \( \log_2 2 = 1 \), and at \( x = 4 \), \( \log_2 4 = 2 \), showing that the function increases as \( x \) grows.
- For \( 0 < x < 1 \), \( \log_2 x \) is negative. For example, \( \log_2 \frac{1}{2} = -1 \) and \( \log_2 \frac{1}{4} = -2 \), as \( 2^{-1} = \frac{1}{2} \) and \( 2^{-2} = \frac{1}{4} \).
- As \( x \to \infty \), \( \log_2 x \to \infty \), since \( 2^y \to \infty \) as \( y \to \infty \).
- As \( x \to 0 \), \( \log_2 x \to -\infty \), since \( 2^y \to 0 \) as \( y \to -\infty \).
Example 2: \( f_2(x) = \log_{1/2} x \)
The function \( f_2(x) = \log_{1/2} x \) is the inverse of \( \left(\frac{1}{2}\right)^x \), so \( y = \log_{1/2} x \iff \left(\frac{1}{2}\right)^y = x \). The behavior is as follows:
- At \( x = 1 \), \( \log_{1/2} 1 = 0 \) because \( \left(\frac{1}{2}\right)^0 = 1 \). Thus, the graph passes through \( (1, 0) \).
- At \( x = \frac{1}{2} \), \( \log_{1/2} \frac{1}{2} = 1 \), and at \( x = \frac{1}{4} \), \( \log_{1/2} \frac{1}{4} = 2 \), indicating the function increases as \( x \to 0^+ \).
- For \( x > 1 \), \( \log_{1/2} x \) is negative. For example, \( \log_{1/2} 2 = -1 \) and \( \log_{1/2} 4 = -2 \), since \( \left(\frac{1}{2}\right)^{-1} = 2 \) and \( \left(\frac{1}{2}\right)^{-2} = 4 \).
- As \( x \to \infty \), \( \log_{1/2} x \to -\infty \), since \( \left(\frac{1}{2}\right)^y \to \infty \) as \( y \to -\infty \).
- As \( x \to 0 \), \( \log_{1/2} x \to \infty \), since \( \left(\frac{1}{2}\right)^y \to 0 \) as \( y \to \infty \).
Observations from Examples
The examples highlight the following important characteristics of logarithmic functions:
-
Domain and Range:
Logarithmic functions are defined only for \( x > 0 \), as the base \( a^y \) is positive for all real \( y \). Hence, the domain is \( (0, \infty) \). The range of \( f(x) = \log_a x \) is \( (-\infty, \infty) \), as \( a^y \) can produce any positive number.
-
Intercept:
The graph of \( f(x) = \log_a x \) intersects the \( x \)-axis at \( (1, 0) \), since \( \log_a 1 = 0 \) for all \( a > 0 \).
-
End Behavior:
For \( a > 1 \), as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to 0 \), \( f(x) \to -\infty \). For \( 0 < a < 1 \), as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to 0 \), \( f(x) \to \infty \).
-
Monotonicity:
If \( a > 1 \), \( f(x) = \log_a x \) is strictly increasing, while if \( 0 < a < 1 \), it is strictly decreasing.
Understanding logarithm as the Inverse of Exponentiation
The logarithmic function \( \log_a x \) is defined as the number \( y \) such that the base \( a \) raised to \( y \) equals \( x \). In mathematical terms:
This relationship expresses that \( \log_a x \) is the inverse operation of raising \( a \) to a power. Just as subtraction undoes addition and division undoes multiplication, the logarithm undoes exponentiation.
Example 1: \( \log_2 8 = 3 \)
Consider \( a = 2 \) and \( x = 8 \). By the definition of the logarithm:
To determine \( y \), solve \( 2^y = 8 \). Recognizing that \( 8 = 2^3 \), it follows that \( y = 3 \). Thus, \( \log_2 8 = 3 \).
This means that \( \log_2 8 \) answers the question: To what power must 2 be raised to obtain 8? The answer is \( 3 \), because \( 2^3 = 8 \).
Example 2: \( \log_3 81 = 4 \)
For \( a = 3 \) and \( x = 81 \), the logarithm \( \log_3 81 \) satisfies:
Since \( 81 = 3^4 \), it follows that \( y = 4 \). Hence, \( \log_3 81 = 4 \). This means that raising \( 3 \) to the power \( 4 \) gives \( 81 \).
Example 3: \( \log_{10} 1000 = 3 \)
In the case of base \( 10 \), the logarithm \( \log_{10} 1000 \) answers:
Since \( 1000 = 10^3 \), it follows that \( y = 3 \), so \( \log_{10} 1000 = 3 \). This shows that \( 10 \) must be raised to the power \( 3 \) to yield \( 1000 \).
The Greatest Integer Function
The greatest integer function \( f(x) = [x] \) assigns to each real number \( x \) the largest integer less than or equal to \( x \).
Definition of the Function
Let \( x \) be a real number. There exists a unique integer \( n \) such that:
The value of the greatest integer function \( [x] \) is defined as \( n \). For example:
- If \( x = 4.8 \), then \( [4.8] = 4 \), because \( 4 \leq 4.8 < 5 \).
- If \( x = -1.3 \), then \( [-1.3] = -2 \), because \( -2 \leq -1.3 < -1 \).
- If \( x = 3 \), then \( [3] = 3 \), as \( x \) is already an integer.
This function captures the idea of identifying the largest integer less than or equal to any real number.
Piecewise Representation
The function \( f(x) = [x] \) can be expressed in piecewise form by describing its behavior over intervals:
Within each interval \( [n, n+1) \), where \( n \in \mathbb{Z} \), the value of \( f(x) = [x] \) is constant and equal to \( n \). The function shifts to the next integer as \( x \) crosses \( n+1 \).
Domain and Range
The domain of \( f(x) = [x] \) includes all real numbers \( x \), as the function is defined for any \( x \). Thus:
The range of the function consists of all integers \( n \), since \( [x] \) only outputs integer values. Hence:
Graph of \( f(x) = [x] \)
The graph of \( f(x) = [x] \) has a characteristic step-like appearance:
- For each interval \( [n, n+1) \), the graph is a horizontal line segment at height \( n \).
- At the left endpoint \( x = n \), the function value is \( n \), represented by a closed circle \( (n, n) \).
- At the right endpoint \( x = n+1 \), the function value "jumps" to \( n+1 \), leaving an open circle \( (n+1, n) \).
For example:
- From \( -2 \leq x < -1 \), \( f(x) = -2 \), and the graph is a horizontal segment from \( (-2, -2) \) to \( (-1, -2) \).
- From \( 1 \leq x < 2 \), \( f(x) = 1 \), and the graph is a horizontal segment from \( (1, 1) \) to \( (2, 1) \).
The graph progresses as a series of such steps, with each step corresponding to a unique integer \( n \).
Behavior of the Function
The greatest integer function is constant within intervals of length 1, and the value shifts by 1 as \( x \) crosses an integer. This results in the following:
- For Positive \( x \): The function rounds \( x \) down to the nearest integer below it. For example, \( [2.7] = 2 \).
- For Negative \( x \): The function still rounds \( x \) down, but for negative values, this means moving further away from zero. For instance, \( [-2.3] = -3 \).
- At Integers: If \( x \) is already an integer, \( [x] = x \), such as \( [5] = 5 \).
Fractional Part Function
The fractional part function, denoted by \( \{x\} \), assigns to a real number \( x \) its fractional part. This is defined as the difference between \( x \) and the greatest integer less than or equal to \( x \), i.e.,
where \( [x] \) is the greatest integer function.
Understanding the Definition
The fractional part \( \{x\} \) represents the "non-integer" portion of \( x \). For example:
- If \( x = 4.7 \), then \( [x] = 4 \), so \( \{x\} = 4.7 - 4 = 0.7 \).
- If \( x = -2.3 \), then \( [x] = -3 \), so \( \{x\} = -2.3 - (-3) = 0.7 \).
- If \( x = 5 \), then \( [x] = 5 \), so \( \{x\} = 5 - 5 = 0 \).
The fractional part \( \{x\} \) satisfies the following properties:
- \( 0 \leq \{x\} < 1 \), regardless of whether \( x \) is positive or negative.
- \( \{x\} = 0 \) if and only if \( x \) is an integer.
Piecewise Representation
The function \( \{x\} = x - [x] \) can be expressed in a piecewise form using the behavior of the greatest integer function \( [x] \). For \( n \in \mathbb{Z} \), the fractional part \( \{x\} \) behaves as:
In general, for \( n \leq x < n+1 \), where \( n \in \mathbb{Z} \), the fractional part is given by:
This formula holds because \( [x] = n \) in this interval, and \( \{x\} = x - [x] \).
Graph of \( \{x\} \)
The graph of \( \{x\} = x - [x] \) consists of line segments within each interval \( [n, n+1) \), where \( n \in \mathbb{Z} \). Key features include:
-
Within Each Interval \( [n, n+1) \):
- The function starts at \( \{n\} = n - n = 0 \) (closed point).
- The function increases linearly with slope 1.
- At \( x = n+1 \), the function reaches \( \{n+1\} = n+1 - n = 1 \) (open point).
-
Periodic Structure:
- The function repeats the same linear pattern over each interval.
- The graph consists of a series of identical segments, each of length 1, spanning from \( y = 0 \) to \( y = 1 \).
For example:
- On \( [0, 1) \), \( \{x\} = x \), so the graph is a line from \( (0, 0) \) (closed) to \( (1, 1) \) (open).
- On \( [1, 2) \), \( \{x\} = x - 1 \), so the graph is a line from \( (1, 0) \) (closed) to \( (2, 1) \) (open).
- On \( [-1, 0) \), \( \{x\} = x - (-1) = x + 1 \), so the graph is a line from \( (-1, 0) \) (closed) to \( (0, 1) \) (open).
Domain and Range
-
Domain:
The function \( \{x\} \) is defined for all real numbers \( x \), as \( [x] \) is defined for all \( x \). Hence:
\[ \text{Domain} = (-\infty, \infty). \] -
Range:
The function \( \{x\} \) always satisfies \( 0 \leq \{x\} < 1 \), as it represents the fractional part of \( x \). Hence:
\[ \text{Range} = [0, 1). \]
Behavior of the Function
The fractional part function \( \{x\} = x - [x] \) isolates the "non-integer" part of \( x \). It is periodic with period 1, as the behavior within any interval \( [n, n+1) \) repeats for all \( n \in \mathbb{Z} \). The function is piecewise linear with a slope of 1 on each interval and resets to 0 at every integer value of \( x \).
Trigonometric Functions
For more description, refer to trigonometry.
-
\( \sin x \):
- The graph of \( \sin x \) is a smooth, periodic wave oscillating between \( -1 \) and \( 1 \).
- The function has a period of \( 2\pi \), meaning it repeats every \( 2\pi \).
- The function crosses the \( x \)-axis at \( x = n\pi, \, n \in \mathbb{Z} \).
-
\( \cos x \):
- The graph of \( \cos x \) is similar to \( \sin x \), but it starts at \( y = 1 \) when \( x = 0 \).
- Like \( \sin x \), it oscillates between \( -1 \) and \( 1 \) with a period of \( 2\pi \).
- The function crosses the \( x \)-axis at \( x = \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z} \).
-
\( \tan x \):
- The graph of \( \tan x \) has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z} \), where the function is undefined.
- The function is periodic with period \( \pi \) and has a range of \( (-\infty, \infty) \).
- The graph crosses the \( x \)-axis at \( x = n\pi, \, n \in \mathbb{Z} \).
-
\( \cot x \):
- The graph of \( \cot x \) has vertical asymptotes at \( x = n\pi, \, n \in \mathbb{Z} \), where the function is undefined.
- It is periodic with period \( \pi \) and has a range of \( (-\infty, \infty) \).
- The function decreases and crosses the \( x \)-axis at \( x = \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z} \).
-
\( \sec x \):
- The graph of \( \sec x \) is derived from \( \cos x \) and is undefined at \( x = \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z} \).
- It has a periodicity of \( 2\pi \), and its range is \( (-\infty, -1] \cup [1, \infty) \).
- The function forms a series of U-shaped curves above \( y = 1 \) and below \( y = -1 \).
-
\( \csc x \):
- The graph of \( \csc x \) is derived from \( \sin x \) and is undefined at \( x = n\pi, \, n \in \mathbb{Z} \).
- It has a periodicity of \( 2\pi \), and its range is \( (-\infty, -1] \cup [1, \infty) \).
- The function forms a series of U-shaped curves similar to \( \sec x \) but centered on \( x = n\pi + \frac{\pi}{2} \).
Below is a table summarizing the domain and range of the six primary trigonometric functions.
| Function | Domain | Range |
|---|---|---|
| \( \sin x \) | \( (-\infty, \infty) \) | \( [-1, 1] \) |
| \( \cos x \) | \( (-\infty, \infty) \) | \( [-1, 1] \) |
| \( \tan x \) | \( x \neq \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z} \) | \( (-\infty, \infty) \) |
| \( \cot x \) | \( x \neq n\pi, \, n \in \mathbb{Z} \) | \( (-\infty, \infty) \) |
| \( \sec x \) | \( x \neq \frac{\pi}{2} + n\pi, \, n \in \mathbb{Z} \) | \( (-\infty, -1] \cup [1, \infty) \) |
| \( \csc x \) | \( x \neq n\pi, \, n \in \mathbb{Z} \) | \( (-\infty, -1] \cup [1, \infty) \) |
Domain and Range of Inverse Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| \( \sin^{-1} x \) | \( [-1, 1] \) | \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) |
| \( \cos^{-1} x \) | \( [-1, 1] \) | \( [0, \pi] \) |
| \( \tan^{-1} x \) | \( (-\infty, \infty) \) | \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) |
| \( \cot^{-1} x \) | \( (-\infty, \infty) \) | \( (0, \pi) \) |
| \( \sec^{-1} x \) | \( (-\infty, -1] \cup [1, \infty) \) | \( \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \) |
| \( \csc^{-1} x \) | \( (-\infty, -1] \cup [1, \infty) \) | \( \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right] \) |
Each domain represents the valid input values for the corresponding inverse function, and each range is the principal value interval chosen for its output to ensure the function is well-defined and single-valued.
Min and Max Function
Let \( f(x) \) and \( g(x) \) be two functions defined on domains \( D_f \) and \( D_g \), respectively. The maximum of \( f(x) \) and \( g(x) \), denoted as \( \max[f(x), g(x)] \), is defined as the larger of the two values at each point \( x \) in \( D_f \cap D_g \). Mathematically,
Similarly, the minimum of \( f(x) \) and \( g(x) \), denoted as \( \min[f(x), g(x)] \), is the smaller of the two values at each point \( x \) in \( D_f \cap D_g \). It is defined as
The domain of both \( \max[f(x), g(x)] \) and \( \min[f(x), g(x)] \) is \( D_f \cap D_g \), as the definitions require both \( f(x) \) and \( g(x) \) to be defined at each \( x \).
The definitions for \( \max[f(x), g(x)] \) and \( \min[f(x), g(x)] \) can also be expressed using algebraic formulas that do not rely on conditions. Specifically,
To understand why these formulas are true, consider the properties of absolute values. For any two real numbers \( a \) and \( b \), we have
Using this, we can rewrite \( f(x) + g(x) \) and \( |f(x) - g(x)| \) as follows:
-
If \( f(x) \geq g(x) \), then \( |f(x) - g(x)| = f(x) - g(x) \). Substituting into the formula for \( \max[f(x), g(x)] \),
\[ \max[f(x), g(x)] = \frac{f(x) + g(x)}{2} + \frac{f(x) - g(x)}{2} = f(x). \]Similarly, for \( \min[f(x), g(x)] \),
\[ \min[f(x), g(x)] = \frac{f(x) + g(x)}{2} - \frac{f(x) - g(x)}{2} = g(x). \] -
If \( f(x) < g(x) \), then \( |f(x) - g(x)| = g(x) - f(x) \). Substituting into the formula for \( \max[f(x), g(x)] \),
\[ \max[f(x), g(x)] = \frac{f(x) + g(x)}{2} + \frac{g(x) - f(x)}{2} = g(x). \]Similarly, for \( \min[f(x), g(x)] \),
\[ \min[f(x), g(x)] = \frac{f(x) + g(x)}{2} - \frac{g(x) - f(x)}{2} = f(x). \]
Thus, the algebraic formulas for \( \max[f(x), g(x)] \) and \( \min[f(x), g(x)] \) correctly reproduce the piecewise definitions in all cases. These formulas not only simplify the computation but also highlight the symmetric relationship between \( \max \) and \( \min \). For any \( f(x) \) and \( g(x) \), we have the identity
which follows directly from their definitions and reflects the complementary nature of the maximum and minimum operations.
The concept of minimum and maximum functions can be extended to \( n \) functions \( f_1(x), f_2(x), \ldots, f_n(x) \), defined on a common domain \( D = \bigcap_{i=1}^n D_{f_i} \), where \( D_{f_i} \) is the domain of \( f_i(x) \).
For the minimum function of \( n \) functions, we define:
Similarly, the maximum function is defined as:
Handling Multiple Functions
For computational purposes, the above definitions can be expressed iteratively. Let:
Graphical Approach to Solving Min/Max Problems
The most intuitive and effective way to handle problems involving \( \min \) or \( \max \) functions is through their graphs:
-
Plot the graphs of all functions \( f_1(x), f_2(x), \ldots, f_n(x) \):
- Identify regions where one function dominates (is larger or smaller than the others).
- For the maximum function, select the curve that lies above all others at each point.
- For the minimum function, select the curve that lies below all others at each point.
-
Locate intersection points:
- Intersection points are key to determining the boundaries where one function takes over another in \( \min \) or \( \max \). At these points, \( f_i(x) = f_j(x) \) for some \( i, j \).
- Solve \( f_i(x) = f_j(x) \) algebraically to find the exact boundaries.
-
Segment the domain:
- Divide the domain into intervals based on the intersection points.
- Within each interval, identify the function that dominates for \( \max \) or the one that is dominated for \( \min \).
Example
Given the functions \( f_1(x) = x^2 \), \( f_2(x) = 4 - x^2 \), and \( f_3(x) = x \), determine the piecewise definition of the minimum function \( \min\{f_1(x), f_2(x), f_3(x)\} \).
Solution:
To solve this problem, observe the graph of \( f_1(x) \), \( f_2(x) \), and \( f_3(x) \), and calculate their intersection points to divide the domain into intervals. We need only the abscissas of these intersection points:
-
\( f_1(x) = f_2(x) \):
\[ x^2 = 4 - x^2 \implies 2x^2 = 4 \implies x = \pm\sqrt{2}. \] -
\( f_1(x) = f_3(x) \):
\[ x^2 = x \implies x(x - 1) = 0 \implies x = 0, 1. \] -
\( f_2(x) = f_3(x) \):
\[ 4 - x^2 = x \implies x^2 + x - 4 = 0 \implies x = \frac{-1 \pm \sqrt{17}}{2}. \]The solutions are \( x = \frac{-1 - \sqrt{17}}{2} \) and \( x = \frac{-1 + \sqrt{17}}{2} \).
Now, using the graph and the calculated intersection points, we determine which function is the minimum in each interval.
-
For \( x \in (-\infty, \frac{-1 - \sqrt{17}}{2}] \), \( f_2(x) = 4 - x^2 \) is the minimum function since it lies below \( f_1(x) \) and \( f_3(x) \).
-
For \( x \in [\frac{-1 - \sqrt{17}}{2}, 0] \), \( f_3(x) = x \) is the minimum function because it lies below \( f_1(x) \) and \( f_2(x) \).
-
For \( x \in [0, 1] \), \( f_1(x) = x^2 \) is the minimum function since it lies below \( f_2(x) \) and \( f_3(x) \).
-
For \( x \in [1, \frac{-1 + \sqrt{17}}{2}] \), \( f_3(x) = x \) is the minimum function as it lies below \( f_1(x) \) and \( f_2(x) \).
-
For \( x \in [\frac{-1 + \sqrt{17}}{2}, \infty) \), \( f_2(x) = 4 - x^2 \) is the minimum function because it lies below \( f_1(x) \) and \( f_3(x) \).
Thus, the piecewise definition of the minimum function is: