Let and perform the

indicated operation.

4)( 2 xxf 2)( xxg

1. 2.

3. 4.

)()( xgxf )()( xgxf

)(

)(

xg

xf))(( xgf

Warm-up 3.3

1. 2.

3. 4.

2

2

2

4 2

4

2

2

x x

x x

x x

3

2

2

4 2

2 4 8x x

x x

x

2 4

22 2

22

x

xx x

xx

2

2

2

2

2

4

4

4 4 4

f x

x

x x

x x

Inverse Functions3.3B

Standard: MM2A5 abcd

Essential Question: How do I graph and analyze exponential functions and their inverses?

Vocabulary• Inverse relation – A relation that interchanges the

input and output value of the original relation

• Inverse functions – The original relation and its inverse relation whenever both relations are functions

• nth root of a – b is an nth root of a if bn = a

• Horizontal Line Test – The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once

Graph y = 2x – 4 using a table and a red pencil.

Graph y = ½x + 2 using a table and a blue pencil.

x y x y

Example 1:

0 -4

1 -2

2 0

3 2

-4 0

-2 1

0 2

2 3

4 4 4 4

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

x y x y

a. What do you notice about the two tables?

The input (x) and output (y) are interchanged

0 -4

1 -2

2 0

3 2

-4 0

-2 1

0 2

2 3

4 4 4 4

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

y = x

b. With your pencil, draw the line y = x.

What is the relationship between the red and blue lines and the line y = x?

The line y = x is the line of reflection for the graphs of the red and blue lines.

We say that y = 2x – 4 and y = ½x + 2 are inverse functions.

Let f(x) = 2x – 4 and f-1(x) = ½x + 2 .

To verify that f(x) = 2x – 4 and are inverse functions you must show thatf(f-1(x)) = f-1(f(x)) = x.

1 1( ) 2

2f x x

1( ( ))f f x 1( ( ))f f x

= f(½x + 2)= 2 (½x + 2) – 4= x + 4 – 4 = x

= f-1(2x – 4)= ½(2x – 4) + 2= x – 2 + 2 = x

(2). Using composition of functions, determine if

f(x) = 3x + 1 and g(x) = ⅓x – 1 are inverse

functions?

f(g(x)) = f(⅓x – 1)

= 3(⅓x – 1) + 1

= x – 3 + 1

= x – 2

NO!

Graph y = x2 for x 0 using a table and a red pencil.

Graph y = √x using a table and a blue pencil.

x y x y

Example 3:

0 0

1 1

2 4

3 9

0 0

1 1

4 2

9 3

4 16 16 4

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

x y x y

0 0

1 1

2 4

3 9

0 0

1 1

4 2

9 3

4 16 16 4

a. What do you notice about the two tables?

The input (x) and output (y) are interchanged

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

b. With your pencil, draw the line y = x.

What is the relationship between the red and blue graphs and the line y = x?

The line y = x is the line of reflection for the graphs of the red and blue graphs.

We say that y = x2 and y = √x are inverse functions.

Let f(x) = x2 and f-1(x) = √x .

To verify that f(x) = x2 and are inverse functions you must show thatf(f-1(x)) = f-1(f(x)) = x.

1( ( ))f f x 1( ( ))f f x

=

=

= x

=

=

= x

xxf )(1

)( xf

2)( x

)( 21 xf

2x

To find the inverse of a function that is one-toone, interchange x with y and y with x, then solvefor y.

Find the inverse: 4. y = 3x + 5

Inverse: x = 3y + 5

yx 3

5

3

1x – 5 = 3y

3

5

3

1)(1 xxf

Find the inverse: 5. y = 2x2 – 1

Inverse: x = 2y2 – 1

2

2

1y

x

x + 1 = 2y2

2

1)(1

xxg

yx

2

1

Note: The function y = x2 is not a one-to-one function.

If the input and output were interchanged, thegraph of the new relation would NOT be a function.

y = x2 x = y2

So, we must restrict thedomain of functions that are not one-to-one in order to create a function with an inverse!

Sketch the graph of the inverse of each function.

6). x y

6 3

0 2

x y

-4 -3

0 2

Sketch the graph of the inverse of each function.

7). x y

-2 3

2 2

x y

6 -5

2 2