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Ph.D. Math, UC Berkeley

 

Home >> Pre-Calculus >> 2. Polynomial and Ratioinal Functions

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2.7. Rational Functions and Their Graphs

Thus far we have looked at graphs of polynomials as well as some simple rational functions. In this section, we will investigate graphs of rational functions in more detail.

As we know, rational functions have the general form f(x) = p(x) / q(x), where p and q are polynomials. Whenever q(x) is zero and p(x) is not, f(x) is undefined. This results in a singularity of f(x). For values of x near the singular value, |f(x)| becomes very large. Thus, when graphing rational functions, it is a good idea to first find its singularities.

Example 1: Graph the function f(x) = 1 / (1 - x2). (We already looked at this function in Section 1.6, but now we take a closer look.)

Solution: Before trying to graph f, we compute its singularities. The denominator is 1 - x2, which factors as (1 + x)(1 - x), which has zeros at x=±1. Since the numerator of f is never zero, we see that these values must be the singularities of f. We draw vertical dashed lines, known as asymptotes, at these values of x as shown below:

asymptotes of f(x) = 1 / (1 - x^2)

Next we make a table of values of f(x) versus values of x. In general, we should use several values of x near the singular values. The values we used in constructing the table in Example 3 of Section 1.6 work quite well. Below we reproduce the table and the resulting graph, along with the asymptotes.

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

graph of f(x) = 1 / (1 - x^2)

 

Example 2: Graph the function f(x) = (x2 - 4) / (3x2 + 8x - 3).

Solution: By trial and error, we find that the denominator factors as 3x2 + 8x - 3 = (x + 3)(3x - 1), which has zeros at x = -3 and x = 1/3, while the numerator has zeros at x = ±2. Thus we compute f(x) for values of x from -4 to 4, with more values near x = -1/3 and x = 3. A table of these values is given below, followed by the resulting graph.

x f(x)   x f(x)
-4.0
0.92
 
-0.5
0.60
-3.5
1.43
 
0.0
1.33
-3.4
1.69
 
0.1
1.84
-3.3
2.11
 
0.2
3.09
-3.2
2.94
 
0.3
11.85
-3.1
5.45
 
0.3
-
-3.0
-
 
0.4
-5.65
-2.9
-4.55
 
0.5
-2.14
-2.8
-2.04
 
1.0
-0.38
-2.7
-1.21
 
1.5
-0.11
-2.6
-0.78
 
2.0
0.00
-2.5
-0.53
 
2.5
0.06
-2.0
0.00
 
3.0
0.10
-1.5
0.21
 
3.5
0.13
-1.0
0.38
 
4.0
0.16

graph of f(x) = (x^2 - 4) / (3x^2 + 8x - 3)

 

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