Intermediate Algebra (11th edition)

390 CHAPTER 7 Rational Expressions and Functions

NOTE In general, if the y-values of a rational function approach or as the

x-values approach a real number a, the vertical line is a vertical asymptote of the

graph. Also, if the x-values approach a real number bas increases without bound,

the horizontal liney= bis a horizontal asymptote of the graph.

|x|

x=a

q -q

Graphing a Rational Function

Graph, and give the equations of the vertical and horizontal asymptotes.

Some ordered pairs that belong to the function are listed in the table.

g 1 x 2 =

- 2

x- 3

EXAMPLE 6

–3 3

3

–3

–3 3

3

–3

FIGURE 4

FIGURE 3

x 0 1 2 2.5 2.75 3.25 3.5 4 5 6 7

y 52 12 32 12 4 8- 8 - 4 - 2 - 1 - 32 - 21

• 2 - 1

There is no point on the graph, shown in FIGURE 3,

for because 3 is excluded from the do-

main of the rational function. The dashed line

represents the vertical asymptote and is

not part of the graph. The graph gets closer to the

vertical asymptote as the x-values get closer to 3.

Again, y= 0 is a horizontal asymptote.

x= 3

x=3,

x 3

x

y

• 2
  
  
  • 4
  
  
  • 8
  
  
  
  

4

8

0 2 46

g(x) x––^2 3

NOW TRY

We can solve rational equations with a graphing calculator by finding the x-intercepts

of the graph of the corresponding rational function.

FIGURE 4shows two views of the graph of the following rational function.

The x-intercepts (or zeros) of the graph determine the solution set of

1

x^2

+

1

x

-

3

2

= 0

ƒ 1 x 2 = 0.

ƒ 1 x 2 =

1

x^2

+

1

x

-

3

2

CONNECTIONS

NOW TRY
EXERCISE 6
Graph, and give the equations
of the vertical and horizontal
asymptotes.

ƒ 1 x 2 =

1

x+ 1

NOW TRY ANSWER

6. vertical asymptote: ;
   horizontal asymptote:

x

y

0

f(x) =x + 1^1

–1 1

y= 0

x=- 1

The bottom of each calculator screen gives approximations of the x-values for

which , here These are the solutions of the equa-

tion. Thus, the solution set is 5 - 0.5485838, 1. 2152504 6.

y= 0 - 0.5485838 and 1. 2152504.