Download Reteach Functions and Their Inverses

Name
Date
LESSON
9-5
Class
Reteach
Functions and Their Inverses
Not all relations are functions and not all relations have inverse functions. To decide whether
the inverse of a relation is a function, use the horizontal-line test.
If any horizontal line passes through more than one point on the graph of a relation, the
inverse of the relation is not a function.
Y
Y
X
X
Can you draw a horizontal line that
passes through more than one point
on the graph?
Can you draw a horizontal line that passes
through more than one point on the graph?
No. So the inverse of the relation is a
function.
Yes. So the inverse of the relation is
NOT a function.
Use the horizontal-line test to determine whether the inverse of each
relation is a function.
1.
2.
Y
Y
X
Not a function
3.
X
Function
4.
Y
Y
X
Function
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c09-5_rt.indd 38
X
Not a function
38
Holt Algebra 2
1/4/06 7:05:48 PM
Process Black
Name
LESSON
9-5
Date
Class
Reteach
Functions and Their Inverses (continued)
The inverse of a function, f x , is denoted f 1 x .
To find the inverse of a function f x , write y f x . Then switch x and y and solve the
equation for y.
2
Find the inverse of f x x 1.
Step 1 Write y f x .
2
yx 1
Step 2 Switch x and y.
2
xy 1
Step 3 Solve for y.
Isolate the variable, y.
2
xy 1
Take the square root of both sides.
x 1 y2
x 1 2
y
The domain is {x | x 1}.
x 1 y
Remember, x 1 can be either positive or negative.
Because y could have 2 different x-values, x 1 and x 1 , this inverse is not
a function. Check by graphing the original function on a graphing calculator and use the
horizontal-line test.
The domain of the inverse is the range of f x : { x | x 1 }.
The range of the inverse is the domain of f x : all real numbers.
Find the inverse of each function and state its domain and range.
3
5. f x x
y
x
3
y 2x 1
x y3
x 2y 1
x
f
1
6. f x 2x 1
3 y
x x 1 2y
1
x1
f x _____
2
domain: all real numbers;
range: all real numbers
x 3 2
domain: all real numbers x such that
x 3; range: nonnegative real numbers
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c09-5_rt.indd 39
39
Holt Algebra 2
1/4/06 7:05:54 PM
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Holt Algebra 2
5/16/06 4:08:40 PM