9-5
9-5 Functions
Functions and
and Their
Their Inverses
Inverses
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra2
Holt
Algebra
2
9-5
Functions and Their Inverses
Warm Up
Solve for x in terms of y.
1.
2.
3.
4. y = 2ln x
Holt Algebra 2
9-5
Functions and Their Inverses
Objectives
Graph and recognize inverses of
relations and functions.
Find inverses of functions.
Determine whether the inverse of a
function is a function.
Write rules for the inverses of functions.
Holt Algebra 2
9-5
Functions and Their Inverses
Vocabulary
inverse function
inverse relation
one-to-one function
Holt Algebra 2
9-5
Functions and Their Inverses
You have seen the word inverse used in various
ways.
The additive inverse of 3 is –3.
The multiplicative inverse of 5 is
Holt Algebra 2
9-5
Functions and Their Inverses
You can also find and apply inverses to relations
and functions. To graph the inverse relation,
you can reflect each point across the line y = x.
This is equivalent to switching the x- and yvalues in each ordered pair of the relation.
Remember!
A relation is a set of ordered pairs. A function is a
relation in which each x-value has, at most, one
y-value paired with it.
Holt Algebra 2
9-5
Functions and Their Inverses
Example 1: Graphing Inverse Relations
Graph the relation and
connect the points. Then
graph the inverse. Identify
the domain and range of each
relation.
Graph each ordered pair and
connect them.
Switch the x- and y-values in
each ordered pair.
x
y
Holt Algebra 2
2
0
5
1
6
5
9
8
x
0 1 5
8
y
2 5 6
9
•
•
•
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•
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•
•
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•
●
•
9-5
Functions and Their Inverses
Check It Out! Example 1
Graph the relation and
connect the points. Then
graph the inverse.
Identify the domain and
range of each relation.
Graph each ordered pair and
connect them.
x
1
3
4
5
6
y
0
1
2
3
5
Switch the x- and y-values
in each ordered pair.
x
y
Holt Algebra 2
0
1
1
3
2
4
3
5
5
6
•
•
•
•
•
•
•
•
•
•
9-5
Functions and Their Inverses
When the relation is also a function, you can write the
inverse of the function f(x) as f–1(x). This notation
does not indicate a reciprocal.
Functions that undo each other are inverse functions.
To find the inverse function, use the inverse
operation. In the example above, 6 is added to x in
f(x), so 6 is subtracted to find f–1(x).
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Example 2: Writing Rules for inverses
Find the inverse of
. 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.
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Example 2 Continued
Because the inverse is a function,
The domain of the inverse is the range of
f(x):{x|x  R}.
The range is the domain of f(x):{y|y  R}.
Holt Algebra 2
.
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Check It Out! Example 2 Continued
Step 1 Find the inverse.
3
y=x –2
3
Rewrite the function using y instead of f(x).
x=y –2
Switch x and y in the equation.
x + 2 = y3
Add 2 to both sides of the equation.
3
x + 2 = 3 y3
3
x+2=y
Holt Algebra 2
Take the cube root of both
sides.
Simplify.
9-5
Functions and Their Inverses
Check It Out! Example 2 Continued
Because the inverse is a function,
.
The domain of the inverse is the range of f(x): R.
The range is the domain of f(x): R.
Holt Algebra 2
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Holt Algebra 2
9-5
Functions and Their Inverses
Example 3: Determining Whether Functions Are
Inverses
Determine by composition whether each pair
of functions are inverses.
f(x) = 3x – 1 and g(x) =
1
3
x+1
Find the composition f(g(x)).
1
f(g(x)) = 3(
3
Holt Algebra 2
x + 1) – 1
Substitute
x in f.
1
3
x + 1 for
= (x + 3) – 1
Use the Distributive
Property.
=x+2
Simplify.
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Example 3B: Determining Whether Functions Are
Inverses
For x ≠ 1 or 0, f(x) =
1
and g(x) =
x–1
1
+ 1.
x
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.
Holt Algebra 2
9-5
Functions and Their Inverses
Example 3B Continued
Check The graphs are
symmetric about
the line y = x for
all x but 0 and 1.
Holt Algebra 2
9-5
Functions and Their Inverses
Check It Out! Example 3a
Determine by composition whether each pair
of functions are inverses.
f(x) = 2
3
3
x + 6 and g(x) =
x–9
2
Find the composition f(g(x)) and g(f(x)).
f(g(x)) =
2
3
(
3
2
x – 9) + 6
g(f(x)) =
3
2
(
2
3
x + 6) – 9
=x–6 +6
=x+9 –9
=x
=x
Because f(g(x)) = g(f(x)) = x, they are inverses.
Holt Algebra 2
9-5
Functions and Their Inverses
Check It Out! Example 3a Continued
Check The graphs are
symmetric about
the line y = x for
all x.
Holt Algebra 2
9-5
Functions and Their Inverses
Check It Out! Example 3b
f(x) = x2 + 5 and
for x ≥ 0
Find the compositions f(g(x)) and g(f(x)).
f(g(x)) =
+5
= x -10 x + 25 +5
= x – 10 x + 30
Holt Algebra 2
Substitute
in f.
Simplify.
for x
9-5
Functions and Their Inverses
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.
Holt Algebra 2
9-5
Functions and Their Inverses
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
Holt Algebra 2
9-5
Functions and Their Inverses
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: {x|x ≥ 4}; R: {all Real Numbers}
Holt Algebra 2
9-5
Functions and Their Inverses
Lesson Quiz: Part III
3. Determine by composition whether f(x) =
3(x – 1)2 and g(x) =
for x ≥ 0.
yes
Holt Algebra 2
+1 are inverses