Inverse Relations & Inverse Functions

Lesson 1 (introduction)S.E. 2A.4C: describe and analyze the

relationship between a function and its inverse

Inverse Relations & Inverse FunctionsIntroduction

Real Life situation -concept of Inverse relation (Warm up)

Suppose you are given the following directions:• From home go left on Route 23 for 5 miles• Turn right onto Orchard St.• Turn left onto Avon Dr. • Tracy’s house is the 5th house on the right.If you start from Tracy’s house, write down the directions to get home:

• From Tracy’s house, go left on Avon Dr.• Turn right on Orchard St.• Turn left on Route 23• You are home!

Inverse Relations & Inverse FunctionsGuiding Question / Math application

• How did you come up with the directions to get home from Tracy’s?

• ANSWER: Worked backward and reversed directions and turns. This is how the concept of “Inverse Relations and Inverse Functions” works.

• Similarly, in Math you can reverse operations to arrive at the inverse of a mathematical rule or relation.

The inverse relation is the set of ordered pairs obtained by reversing the coordinates of each ordered pair. This means that whenever f contains (a,b) f-1(x) contains (b,a).

Determine the Inverse of a Function Defined by a Map, a Table, or a Set of Ordered Pairs

Find the inverse of the following function.

PRACTICE

Finding the Inverse of a Function Defined by a Table

x y

3 1

5 2

9 4

15 7

x y

1 3

2 5

4 9

7 15

Find the inverse of the following function.

PRACTICEFinding the Inverse of a Function Defined by a Table

x y

-3 5

-2 9

-1 2

0 11

Find the inverse of the following function.

Find the inverse of the following function.

f(x) = {(-3, -27), (-2,-8), (-1,-1), (0,0), (1,1), (2,8), (3,27)}

f-1(x) = {(-27, -3), (-8,-2), (-1,-1), (0,0), (1,1), (8,2), (27,3)}

Find the Domain and Range of the Inverse Function

Domain and Range

What is the domain of f(x)?

How does this relate to f-1(x)?

The domain of f is the range of f -1

What is the range of f(x)?

How does this relate to f-1(x)?

The range of f is the domain of f -1

x y

3 1

5 2

9 4

15 7

x y

1 3

2 5

4 9

7 15

f(x) f-1(x)

PRACTICEFinding the Domain and Range of an Inverse Function

The function f(x) is defined below.

f(x) = {(-2, 5), (-1,3), (3,7), (4,12)}

Find the range of f-1(x).

The function g(x) is defined below.

g(x) = {(6, 11), (-2,7), (0,3), (-5,4)}

Find the range of g-1(x).

Graphing with Patty Paper

• Graph y= 2x – 6 on graph paper

• Graph y=x on graph paper

• Trace y= 2x – 6, y=x, the x-axis and the y-axis on a sheet of patty paper

• Fold the patty paper along the line y=x

• Trace the reflection of y= 2x – 6 onto the patty paper

• Trace the reflection of y= 2x – 6 onto the graph paper using a different color pencil or make it dotted/dashed.

Graphing with Patty Paper

1. What is the equation of the reflection (new line)?2. List 3 points that lie on y= 2x – 6 and list the

coordinates of corresponding points on the reflected graph.

3. What do you notice about the coordinates of the corresponding points?

4. What are the domain and range of y= 2x – 6?5. What are the domain and range of the reflection

of y= 2x – 6?

PRACTICE

The function f(x) is graphed below.

Graph the inverse function.

Which of the following is the graph of the function below and its inverse?

A) B)

C) D)

yx

yx

2

22

2

Process for finding the Inverse Function

1. Switch x and y

2. Solve for y

Find the inverse of xy 2

1. Exchange x and yyx 2

2. Solve for y

Example

Find the inverse of the following function. y=2x+51.Switch x and y.

x=2y+52.Solve for y.

x-5=2y (Subtract 5)(x-5)/2=y (Divide by 2)