Intermediate Algebra (11th edition)

10.1 Inverse Functions

Horizontal Line Test

A function is one-to-one if every horizontal line intersects
the graph of the function at most once.

Inverse Functions

For a one-to-one function ƒ defined by an equation
the equation that defines the inverse function
is found by interchanging xand y, solving for y, and
replacing ywith

In general, the graph of is the mirror image of the
graph of ƒ with respect to the line y=x.

ƒ -^1

ƒ -^11 x 2.

ƒ -^1

y=ƒ 1 x 2 ,

Find if

The graph of ƒ is a non-horizontal straight line, so ƒ is one-to-one by

the horizontal line test.

To find interchange xand yin the equation

Solve for yto get

Therefore,

The graphs of a function ƒ and its inverse ƒ-^1 are shown here.

ƒ -^11 x 2 =

x+ 3

2

, or ƒ -^11 x 2 =

1

2

x+

3

2

.

y=

x+ 3

2

.

x= 2 y- 3

ƒ-^11 x 2 , y= 2 x-3.

ƒ -^1 ƒ 1 x 2 = 2 x-3.

QUICK REVIEW

CONCEPTS EXAMPLES

(^01)
y
2
4
f(x) = 3x
x
x
y
4
–5
–5 4
0
y = x
f
f –1

10.2 Exponential Functions

For defines the exponential
function with base a.

Graph of

1.The graph contains the point

2.When the graph rises from left to right. When
the graph falls from left to right.

3.The x-axis is an asymptote.

4.The domain is 1 - q, q 2 , and the range is 1 0, q 2.

06 a 6 1,

a 7 1,

1 0, 1 2.

ƒ 1 x 2 ax

a 7 0,aZ1,ƒ 1 x 2 ax ƒ 1 x 2 = 3 xdefines the exponential function with base 3.

10.3 Logarithmic Functions

means

For andb 7 0, bZ1, logb b 1 logb 1 0.

yloga x xa y. means

log 3 3 = 1 log 5 1 = 0

y=log 2 x x= 2 y.

624 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

(continued)