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David Terr
Ph.D. Math, UC Berkeley

1.6. Graphs of Functions

One nice property of functions is that they are easy to graph. The graph of the function f is the collection of points (x, f(x)), where x ranges over the domain of f. Thus to graph a function f, one merely needs to compute y=f(x) for sufficiently many values of the variable x in the domain of f. We consider several examples below.

Example 1: Graph the function f(x) = x2.

Solution: First we compute several values of f(x) = x2 corresponding to small values of x. The following table lists several values.

x f(x)
-5
25
-4
16
-3
9
-2
4
-1
1
0
0
1
1
2
4
3
9
4
16
5
25

Now it is an easy matter to plot these points and connect them with a smooth curve. The resulting graph is shown below:

Figure 1.6.1: Graph of a Parabola

This is an important type of function, known as a parabola. We will encounter parabolas again in the following chapter.

Example 2: Graph the function f(x) = 1/x.

Solution: As before, we compute f(x) for several values of x. Note that f(x) is undefined when x=0, so this value of x does not lie in the domain of f.

x f(x)
-5
-0.2
-4
-0.25
-3
-0.3
-2
-0.5
-1
-1
-0.5
-2
-0.3
-3
-0.25
-4
-0.2
-5
0
-
0.2
5
0.25
4
0.3
3
0.5
2
1
1
2
0.5
3
0.3
4
0.25
5
0.2

Since f(x) is undefined when x=0, the graph of f consists of two parts, known as branches, one corresponding to negative values of the variable x and the other corresponding to positive values of x. The point x=0 is known as a singularity of f(x). The graph of f(x) is shown below:

Figure 1.6.2: Graph of a Hyperbola

This is another important function, known as a hyperbola, which we will encounter again in Chapter 9.

Example 3: Graph the function f(x) = 1 / (1 - x2).

Solution: As before, we compute f(x) for several values of x. Values of f(x) are rounded to the nearest hundredth. Note that f(x) is undefined when x=±1, so these value of x do not lie in the domain of f.

x f(x)
-2.0
-0.33
-1.8
-0.45
-1.6
-0.64
-1.4
-1.04
-1.2
-2.27
-1.0
-
-0.8
2.78
-0.6
1.56
-0.4
1.19
-0.2
1.04
0.0
1.00
0.2
1.04
0.4
1.19
0.6
1.56
0.8
2.78
1.0
-
1.2
-2.27
1.4
-1.04
1.6
-0.64
1.8
-0.45
2.0
-0.33

Once again, it is straightforward to plot these points and graph the function. Note that f(x) consists of three branches and that like the parabola in Example 1, it is symmetric about the y-axis.