7.7 Inverse Relations 7.7 Inverse Relations and Functionsand Functions

Using a graphing calculator, graph the Using a graphing calculator, graph the pairs of equations on the same graph. pairs of equations on the same graph. Sketch your results. Be sure to use the Sketch your results. Be sure to use the negative sign, not the subtraction key.negative sign, not the subtraction key.

82)8(2/1

xyxy

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” An inverse relation “undoes” the relation and switches the x the relation and switches the x and y coordinates.and y coordinates.

In other words, if the relation has In other words, if the relation has coordinates (a, b), the inverse has coordinates (a, b), the inverse has coordinates of (b,a)coordinates of (b,a)

XX YY00 33

11 44

-3-3 00

-5-5 22

22 55

-8-8 55

XX YY33 00

44 11

00 -3-3

22 -5-5

55 22

55 -8-8

Function f(x) Inverse of Function f(x)

Let’s look at our graphs from earlier. Let’s look at our graphs from earlier. Notice that the points of the graphs are Notice that the points of the graphs are reflected across a specific line. reflected across a specific line.

82)8(2/1

xyxy

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 Finding the Inverse of an equationequation

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 What happens if I don’t include the include the ++ ? ?

Graphing the Graphing the function and only the function and only the positive graph of the positive graph of the inverse . . . inverse . . . We only get

half of the inverse graph.

Finding the Inverse of a Finding the Inverse of a functionfunction

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 SomeLet’s Try Some

2310 xy 7)( xxf

Find the inverse of each

Let’s Try SomeLet’s Try Some

2310 xy 7)( xxf

Find the inverse of each

Composition of Inverse Composition of Inverse FunctionsFunctions

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

thenfunctions, inverse aref and f If1-1-

-1`

xxf

3743741 (374) )f(f so

1 )f(f1-

-1

For the function

)374)(( find ,6

7)( 1- ffxxf