Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

459

Reduce the fraction to lowest terms.

20.34 Simplify the complex fraction: .

Express the complex fraction as a quotient.

Factor the quadratic expressions and simplify the product.

Graphing Rational Functions

Rational functions have asymptotes

20.35 Given a function , describe how to identify the vertical and

horizontal asymptotes of its graph.

If x = c is a vertical asymptote of the graph, then d(c) = 0 but n(c) ≠ 0. To

determine the horizontal asymptotes of f(x), if any exist, consider the degrees

of the numerator and denominator. Let a be the degree of n(x) and b represent

the degree of d(x).

You can’t

cancel out the

x’s here to get

, because x and

–7 aren’t multiplied in

the denominator.

Set the

denominator

equal to zero

and solve. The

solutions represent

vertical asymptotes as

long as they don’t also

make the numerator

equal 0 when you

plug them into

n(x).

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

460

◆ If a > b, then f(x) has no horizontal asymptotes.

◆ If a < b, then the y-axis is the horizontal asymptote of f(x).

◆ If a = b, then the horizontal asymptote of f(x) is

.

Note: Problems 20.36–20.38 refer to the function .

20.36 Identify the vertical asymptote to the graph of g(x).

Set the denominator of g(x) equal to 0 and solve for x.

The line x = –2 is a vertical asymptote to the graph of g(x) because g(–2) is

undeﬁned.

Note: Problems 20.36–20.38 refer to the function .

20.37 Identify the horizontal asymptote to the graph of g(x).

The degree of the numerator of g(x) is 0, and the degree of the denominator is

1. According to Problem 20.35, when the degree of the numerator is less than

the degree of the denominator, the y-axis, g(x) = 0, is the horizontal asymptote

to the graph.

Note: Problems 20.36–20.38 refer to the function .

20.38 Graph g(x).

To transform the function into g(x), you add two to the input of the

function, which shifts the graph of two units to the left. The graph of g(x)

is presented in Figure 20-1.

If the

highest

powers of x in

the numerator

and denominator

are equal, then the

coefcient attached

to the highest power in

the numerator divided

by the coefcient

attached to the

highest power of the

denominator is

the horizontal

asymptote.

A function is

undened when

its denominator

equals zero and its

numerator doesn’t. In

this case,

g(–2) =

.

The highest

power of x in the

denominator is x

1

, so

its degree is 1. There

are no variables in the

numerator, so it has

degree 0.

It’s more correct

to say the equation

of the y-axis is g(x) = 0

i

nstead y = 0, because

technically there aren’t

any y’s in the equation.

Write g(x) where you’d

normally write y.

See Problem

16.34 for more

information.

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

461

Figure 20-1: The graph of has vertical asymptote x = –2 and horizontal

asymptote g(x) = 0.

Note: Problems 20.39–20.41 refer to the function .

20.39 Identify the vertical asymptote(s) to the graph of h(x).

Factor the numerator and denominator of h(x).

Set the factors of the denominator equal to zero and solve for x.

Because is undeﬁned, is a vertical asymptote to the graph of h(x).

If you plug

x = –6 into h(x), it

makes the numerator

and denominator equal

zero. For x = –6 to be a

vertical asymptote, it has

to make the denominator

equal zero but not the

numerator.

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

462

Note: Problems 20.39–20.41 refer to the function .

20.40 Identify the horizontal asymptote to the graph of h(x), if one exists.

The numerator and denominator both have degree two. According to Problem

20.35, the horizontal asymptote is the quotient of the leading coefﬁcients of the

numerator and denominator. Therefore, the graph of h(x) has horizontal

asymptote .

Note: Problems 20.39–20.41 refer to the function .

20.41 Use a table of values to plot the graph of h(x).

Construct a table of values that includes x-values near the vertical asymptote,

because the most dramatic changes in the graph of a rational function occur

near its vertical asymptotes.

The graph of h(x) is presented in Figure 20-2.

Plug the

x’s into the

factored version

of h(x) from Problem

20.39. It makes

evaluating the

functions a little

easier.

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