2.5 Inverses

Warm-up (IN)

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

Solve each equation for y.1. 4x y2. 2 3x y

33. 5yx

14. 13

x y

5. Let 2 4 and 0.5 2.

Find and .

f x x g x x

f g x g f x

14

y x

1 32 2

y x

5 3y x

3 1y x

f g x x

g f x x

Notes

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

Inverse of a function - Domain and range are switched 1f x

Ex 1 – find the inverse of each relation and determine if it’s a function.

a. 1,2 , 4, 2 3,2

inverse 2,1 , 2, 4 2,3 Not a function b. 2, 4 , 3, 4 8, 5

inverse Yes a function

4, 2 , 4,3 5, 8

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

Ex 2 – Find an equation for the inverse of the equation

a. 4 5y x - Switch the x and y, then solve for y!4 5x y -

5 4x y 5

4x y

2 3b. 5xf x

2 3

5yx

5 2 3x y 5 3 2x y

15 32x f x

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

Try it! 1c. 1

4f x x

How can we use inverses in the real world??

1 4 1f x x

How are the graphs of a function and its inverse related?Graph y=x and y=3x-2 and its inverse (use the inverse key) on the same viewing screen. Repeat with y=3x+2, y=-2x+5 and y=x²

**W2L! Explain how the graphs of a function and its inverse are related.

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

Horizontal Line Test – Used to determine if the inverse of a function is a function

Ex 3 – Is the inverse of the graph a function?a. b. c.

yes no yes

Learning Objective: to find the inverse of a relation or function and to determine whether the inverse of a function is a function.

One to One Function –

Passes the horizontal line test, its inverse is also a function.

2 functions are inverses of each other if

f g x x

g f x xand

Ex 4 – 1 3Show that 4 3 and 4 4

are inverses of each other.

f x x g x x

f g x f g x

1 34 4

f x

1 34 34 4x

x 3 3 x

g f x g f x

4 3g x 1 34 34 4

x

x xTry it! Pg. 122 #10 43

43

HW – p. 122 #11,19,21,27,28,30-48(mult. of 3), 51, 52

Out – Describe (in words!) the procedure for finding the inverse of y=4x-1.

Summary – I think I can remember…

POW!!