Copyright © Cengage Learning. All rights reserved.

7Rational Functions

Copyright © Cengage Learning. All rights reserved.

7.1 Rational Functions

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Objectives

Identify a rational function.

Find rational functions that model an application.

Find the domain of a rational function.

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Rational Functions

Rational functions are functions that contain fractions involving polynomials.

These functions can be simple or very complex.

Rational functions often result from combining two functions together using division.

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Rational Functions

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Example 1 – Cost per student

A group of students in the chess club wants to rent a bus to take them to the national chess competition. The bus is going to cost $1500 to rent and can hold up to 60 people.

a. Write a model for the cost per student to rent the bus if s students take the bus and each student pays an equal share.

b. How much would the cost per student be if 30 students take the bus?

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Example 1 – Cost per student

c. How much would the cost per student be if 60 students take the bus?

d. What would a reasonable domain and range be for this model? Explain.

cont’d

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Example 1(a) – Solution

Let C(s) be the cost per student in dollars for s students to take the bus to the national chess competition. Because each student is going to pay an equal amount, we might consider a few simple examples:

If only one student takes the bus, that student would have to pay $1500.

If two students take the bus, they will have to paydollars each.

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Example 1(a) – Solution

We are taking the total cost of $1500 and dividing it by the number of students taking the bus.

This pattern would continue, and we would get the following function.

cont’d

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Example 1(b) – Solution

If 30 students take the bus, we can substitute 30 for s and calculate C(30).

Therefore, if 30 students take the bus, it will cost $50 per

person.

cont’d

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Example 1(c) – Solution

Substituting in s = 60, we get

Therefore, if 60 students take the bus, it will cost $25 per person.

cont’d

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Example 1(d) – Solution

In an application problem, we will continue to avoid model breakdown when setting a domain. Because the bus can hold only up to 60 people, we must limit the domain to positive numbers up to 60.

This means that we could have a possible domain of [1, 60]. With this domain, the range would be [25, 1500].

Of course, there are other possible domains and ranges, but these would be considered reasonable.

cont’d

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Domain of a Rational Function

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Domain of a Rational Function

When considering the domain of a rational function, we will mainly be concerned with excluding values from the domain that would result in the denominator being zero.

The easiest way to determine the domain of a rational function is to set the denominator equal to zero and solve.

The domain then becomes all real numbers except those values that make the denominator equal zero.

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Domain of a Rational Function

Any place where the denominator is zero would result in a vertical asymptote or a hole with a missing value.

The graph of a function will not touch a vertical asymptote but instead will get as close as possible and then jump over it and continue on the other side.

Whenever an input value makes the numerator and denominator both equal to zero, a hole in the graph will occur instead of a vertical asymptote.

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Domain of a Rational Function

Consider the two graphs below to see when a vertical asymptote occurs and when a hole occurs.

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Example 2 – Finding the domain of a rational function

Find the domain of the following rational functions. Determine whether the excluded values represent where a vertical asymptote or a hole appear in the graph.

a. b.

c. d.

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Example 2 – Finding the domain of a rational function

e.

cont’d

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Example 6(a) – Solution

Because the denominator of the function would

be zero when x = 0, we have a domain of all real numbers except zero.

This can also be written simply as x ≠ 0. When x = 0,

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Example 2(a) – Solution

The denominator is zero but the numerator is not, so a vertical asymptote occurs when x = 0.

Looking at the graph of f (x), we see that the function jumps over the input value x = 0, and there is a vertical asymptote in its place.

cont’d

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Example 2(b) – Solution

The denominator of the function would be zero when so its domain is all real numbers such that When

The denominator is zero but the numerator is –4, so there is a vertical asymptote at x = –9.

cont’d

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Example 2(b) – Solution

Looking at this graph again, we see a vertical asymptote. Pay attention to the way in which this function must be entered into the calculator with parentheses around the numerator and another set of parentheses around the denominator of the fraction.

cont’d

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Example 2(c) – Solution

If you set the denominator of the functionequal to zero, you get

(x + 4)(x – 7) = 0

x + 4 = 0 x – 7 = 0

x = –4 x = 7

Therefore, the domain is all real numbers except x = –4 or 7. When x = –4,

cont’d

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Example 2(c) – Solution

Both the numerator and denominator equal zero, so a hole occurs in the graph when x = –4. When x = 7,

The denominator equals zero but the numerator equals 11, so there is a vertical asymptote at x = 7.

cont’d

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Example 2(c) – Solution

This graph is shown in two parts so that you can see the hole that appears at x = –4 and then the asymptote at x = 7. Without setting up two windows, it is almost impossible to see the hole.

cont’d

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Example 2(d) – Solution

Set the denominator of the function equal to zero.

x2 + 5x + 6 = 0

(x + 3)(x + 2) = 0

x + 3 = 0 x + 2 = 0

x = –3 x = –2

cont’d

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Example 2(d) – Solution

Therefore, the domain is all real numbers except x = –3 or –2. For both x = –3 and –2, the denominator equals zero but the numerator does not.

Therefore, there are vertical asymptotes at x = –3 and x = –2. This graph has an interesting shape, but it does have two vertical asymptotes. Again the numerator and denominator of the fraction need parentheses around them to create the graph correctly.

cont’d

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Example 2(e) – Solution

The graph of this function shows a vertical asymptote at about x = 3, so the domain should be all real numbers except x = 3. We cannot see any holes in the given graph.

cont’d