NOTES 7.7

Inverse Relations and Functions

Using a graphing calculator, graph the

pairs of equations on the same graph.

Sketch your results. Be sure to use the

negative sign, not the subtraction key.

82

)8(2/1

xy

xy

3

3

32

xy

xy

xy

2

2

22

xy

xy

xy

What do you notice

about the graphs?

These graphs are

said to be inverses

of each other.

An inverse relation “undoes” the

relation and switches the x and

y coordinates.

In other words, if the relation has

coordinates (a, b), the inverse has

coordinates of (b,a)

X Y

0 3

1 4

-3 0

-5 2

2 5

-8 5

X Y

3 0

4 1

0 -3

2 -5

5 2

5 -8

Function f(x) Inverse of

Function f(x)

Let’s look at our graphs from earlier.

Notice that the points of the graphs are

reflected across a specific line.

82

)8(2/1

xy

xy

3

3

32

xy

xy

xy

2

2

22

xy

xy

xy

What is the equation of

the line of reflection?y = x

Finding the Inverse of an

equation

yx 3

Find the inverse of

y=x2+3

x=y2+3

x – 3 = y2

Switch the x and y

Solve for y

Find the square root

of both sides

What happens if I don’t

include the + ?

Graphing the

function and only the

positive graph of the

inverse . . . We only get

half of the

inverse graph.

Finding the Inverse of a

function

2 xxf

22 yx

22 )2( yx

Rewrite using y

Switch the x and y

Square both sides

Solve for y

When we find the inverse of a

function f(x) we write it as f-1

22 xy

2 xy

2 yx

Find the inverse of

Let’s Try Some

2310 xy 7)( xxf

Find the inverse of each

Composition of Inverse

Functions

x ))(f(f and ))((f

thenfunctions, inverse aref and f If

1-1-

-1`

xxf

3743741 (374) )f(f so

1 )f(f

1-

-1

For the function

)374)(( find ,6

7)( 1- ff

xxf