Graph of a Function
Graph of a Function
Let \( f : A \to B \) be a function, where \( A \) is the domain and \( B \) is the codomain. The graph of \( f \), denoted by \( G \), is defined as the set of all ordered pairs \((x, f(x))\) such that \( x \in A \). Mathematically,
This set \( G \) represents the correspondence between each input \( x \) and its output \( f(x) \).
Plotting the Graph
To visualize the function, we can plot the ordered pairs \((x, f(x))\) on a Cartesian coordinate system. Each point on the plot corresponds to an element of \( G \). This visual representation allows us to observe how the output of the function \( f(x) \) changes with respect to the input \( x \).
Example
Graph of \( f : \mathbb{Z} \to \mathbb{Z} \), \( f(x) = x^2 \)
The function \( f : \mathbb{Z} \to \mathbb{Z} \) is defined by \( f(x) = x^2 \). The graph of \( f \) is the set \( G = \{ (x, x^2) : x \in \mathbb{Z} \} \), which contains infinitely many ordered pairs. Since it is impossible to display all these pairs on paper, we select a finite subset of points for illustration.
Here are some values of \( x \in \mathbb{Z} \) and their corresponding outputs:
The graph of \( f(x) = x^2 \) includes points \((x, x^2)\), where \( x \in \mathbb{Z} \). For this example, we plot the selected points \((-3, 9)\), \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), \((2, 4)\), and \((3, 9)\).
Now, we will visualize the graph.
The graph above shows the points \((-3, 9)\), \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), \((2, 4)\), and \((3, 9)\) plotted for the function \( f(x) = x^2 \). The points are symmetric about the \( y \)-axis.
Continuous curve
Graph of \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = x^2 \)
The function \( f(x) = x^2 \) is a mapping from the set of real numbers \( \mathbb{R} \) to itself, defined by \( f(x) = x^2 \). The graph of this function is the set of points:
Continuous Nature of the Graph
Unlike the case when \( f : \mathbb{Z} \to \mathbb{Z} \), where the graph consisted of discrete points, here the graph is continuous. This is because real numbers have the property that between any two distinct real numbers, there are infinitely many others. Consequently, between any two points \( (x_1, x_1^2) \) and \( (x_2, x_2^2) \) on the graph, there exist infinitely many additional points. This results in a smooth curve rather than isolated points.
Sample Points for Visualization
To construct a part of the graph, consider a few points where \( x \) takes specific values:
Between any two points, such as \( (-2, 4) \) and \( (-1, 1) \), there are infinitely many points like \( (-1.5, 2.25) \), which collectively form a smooth, continuous curve.
Visualization of the Curve
To sketch the graph, begin with the vertex at \( (0, 0) \) and plot symmetric points on either side of the \( y \)-axis:
Example
Graph of \( h : \mathbb{R} \to \mathbb{R} \), \( h(x) = x - x^2 \)
The function \( h(x) = x - x^2 \) is defined as \( h : \mathbb{R} \to \mathbb{R} \). Its graph is given by:
Sample Points:
We evaluate \( h(x) \) at several \( x \)-values and organize them in the table below:
We plot these points on a Cartesian coordinate system. A smooth curve can then be drawn passing through these points since the function \( h(x) \) is continuous.
Understanding the Purpose of Graphs
In the examples so far, we have been plotting graphs by evaluating a function at several sample points and then joining these points to form a curve. This method works well for simple functions, but it becomes impractical for more complicated functions.
Complex functions often have more turns, loops, and sudden jumps. To capture these behaviors accurately, you would need to calculate a large number of sample points. For example, if a function oscillates rapidly or changes direction frequently, missing even a few critical points could lead to an inaccurate graph. You would not want to spend the time and effort evaluating so many points manually.
Why Do We Draw Graphs?
Graphs are essential tools for problem-solving. They provide a quick and visual way to understand the behavior of a function. We use them to:
- Identify trends and patterns.
- Analyze function properties, such as monotonicity and symmetry.
- Locate roots, maxima, and minima.
- Study real-world phenomena modeled by functions.
Since graphs are meant for quick analysis, relying on sample point calculations defeats the purpose. Instead, we need efficient methods to draw graphs that are both accurate and fast.
Beyond Sample Points: Using Calculus for Graphs
As we delve deeper into calculus, we will discover powerful techniques that allow us to draw graphs without relying solely on sample points. These techniques include:
- Finding critical points: Using derivatives to identify where the function increases, decreases, or changes direction.
- Analyzing concavity: Understanding the curvature of the graph through second derivatives.
- Locating asymptotes and discontinuities: Examining the behavior of functions at extreme values or undefined points.
These tools will enable us to construct accurate graphs systematically and efficiently, even for the most complex functions. For now, understanding the limitations of sample-based plotting is a crucial step toward mastering the art of graphing.