Functions and Their Inverses

1
6-6
Functions and Their Inverses
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra2
Holt McDougal Algebra 2
2
Warm Up Solve for x in terms of y.
1.
2.
3.
4. y 2ln x
3
Objectives
Determine whether the inverse of a function is a
function. Write rules for the inverses of
functions.
4
Vocabulary
one-to-one function
5
In Lesson 7-2, you learned that the inverse of a
function f(x) undoes f(x). Its graph is a
reflection across line y x. The inverse may or
not be a function.
Recall that the vertical-line test (Lesson 1-6)
can help you determine whether a relation is a
function. Similarly, the horizontal-line test can
help you determine whether the inverse of a
function is a function.
6
(No Transcript)
7
Example 1A Using the Horizontal-Line Test
Use the horizontal-line test to determine whether
the inverse of the blue relation is a function.
The inverse is a function because no
horizontal line passes through two points on the
graph.
8
Example 1B Using the Horizontal-Line Test
Use the horizontal-line test to determine whether
the inverse of the red relation is a function.
The inverse is a not a function because a
horizontal line passes through more than one
point on the graph.
9
Check It Out! Example 1
Use the horizontal-line test to determine whether
the inverse of each relation is a function.
The inverse is a function because no
horizontal line passes through two points on the
graph.
10
Recall from Lesson 7-2 that to write the rule for
the inverse of a function, you can exchange x and
y and solve the equation for y. Because the value
of x and y are switched, the domain of the
function will be the range of its inverse and
vice versa.
11
Example 2 Writing Rules for inverses
Step 1 The horizontal-line test shows that the
inverse is a function. Note that the domain and
range of f are all real numbers.
12
Example 2 Continued
Step 1 Find the inverse.
Rewrite the function using y instead of f(x).
Switch x and y in the equation.
Cube both sides.
Simplify.
Isolate y.
13
Example 2 Continued
Check Graph both relations to see that they are
symmetric about y x.
14
Check It Out! Example 2
Find the inverse of f(x) x3 2. Determine
whether it is a function, and state its domain
and range.
Step 1 The horizontal-line test shows that the
inverse is a function. Note that the domain and
range of f are all real numbers.
15
Check It Out! Example 2 Continued
Step 1 Find the inverse.
y x3 2
Rewrite the function using y instead of f(x).
x y3 2
Switch x and y in the equation.
Add 2 to both sides of the equation.
x 2 y3
Take the cube root of both sides.
Simplify.
16
Check It Out! Example 2 Continued
The domain of the inverse is the range of f(x) R.
The range is the domain of f(x) R.
Check Graph both relations to see that they are
symmetric about y x.
17
You have seen that the inverses of functions are
not necessarily functions. When both a relation
and its inverses are functions, the relation is
called a one-to-one function. In a one-to-one
function, each y-value is paired with exactly one
x-value.
You can use composition of functions to verify
that two functions are inverses. Because inverse
functions undo each other, when you compose two
inverses the result is the input value x.
18
(No Transcript)
19
Example 3 Determining Whether Functions Are
Inverses
Determine by composition whether each pair of
functions are inverses.
f(x) 3x 1 and g(x) x 1
Find the composition f(g(x)).
Use the Distributive Property.
(x 3) 1
Simplify.
x 2
20
Example 3 Continued
Because f(g(x)) ? x, f and g are not inverses.
There is no need to check g(f(x)).
Check The graphs are not symmetric about the line
y x.
21
Example 3B Determining Whether Functions Are
Inverses
Find the compositions f(g(x)) and g(f (x)).
(x 1) 1
x
x
Because f(g(x)) g(f (x)) x for all x but 0
and 1, f and g are inverses.
22
Example 3B Continued
Check The graphs are symmetric about the line y
x for all x but 0 and 1.
23
Check It Out! Example 3a
Determine by composition whether each pair of
functions are inverses.
Find the composition f(g(x)) and g(f(x)).
x 6 6
x 9 9
x
x
Because f(g(x)) g(f(x)) x, they are inverses.
24
Check It Out! Example 3a Continued
Check The graphs are symmetric about the line y
x for all x.
25
Check It Out! Example 3b
f(x) x2 5 and for x 0
Find the compositions f(g(x)) and g(f(x)).
Simplify.
26
Check It Out! Example 3b Continued
Because f(g(x)) ? x, f and g are not inverses.
There is no need to check g(f(x)).
Check The graphs are not symmetric about the line
y x.
27
Lesson Quiz Part I
1. Use the horizontal-line test to determine
whether the inverse of each relation is a
function.
A yes B no
28
Lesson Quiz Part II
2. Find the inverse f(x) x2 4. Determine
whether it is a function, and state its domain
and range.
not a function
D xx 4 R all Real Numbers
29
Lesson Quiz Part III
3. Determine by composition whether f(x) 3(x
1)2 and g(x) 1 are inverses for x 0.
yes