Today in Pre-Calculus

• Go over homework questions• Notes: Inverse functions• Homework

Inverse Functions• Reversing the x- and y-coordinates of all the ordered

pairs in a relation gives the inverse.• The inverse of a relation is a function if it passes the

horizontal line test.

• A graph that passes both the horizontal and vertical line tests is a one-to-one function. This is because every x is paired with a unique y and every y is paired with a unique x.

Inverse Functions• Definition: If f is a one-to-one function with domain D

and range R, then the inverse function of f, denoted f –1, is the function with domain R and range D defined by f –1(b)=a iff f(a)=b

Graphing Inverses

Examplea) f(x) = 2x – 3

y = 2x – 3 x : (-∞,∞), y: (-∞,∞)

x = 2y – 3 y : (-∞,∞), x: (-∞,∞)

x + 3 = 2y

D: (-∞,∞)1 3

( )2 2

f x x -1

1 3

2 2y x

Example

f(x) =

y = x = [0,∞) , y = [0,∞)

x = y = [0,∞) , x = [0,∞)

y = x2

f –1(x) = x2 D=[0,∞)

x

x

y

Example

x ≠ -2 , y ≠ 1

y ≠-2 , x ≠1

x(y+2) = y xy + 2x = y 2x = y – xy 2x = y(1-x)

( )2

xf x

x

2

xy

x

2

yx

y

2

1

xy

x

1 2( ) : ( ,1) (1, )

1

xf x D

x

Inverse Composition Rulestates that a function f is one-to-one with inverse function g iff f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f.

Used to verify that f and g are inverses of each other.

Example3

( ) 2 - 3 ( )2

xf x x and g x

(2 3) 3 2( ( )) 2 3

2 2

x xg f x g x x

3 3( ( )) 2 3 3 3

2 2

x xf g x f x x

Example2( ) ( )f x x and g x x

2

( ( ))g f x g x x x

2 2( ( ))f g x f x x x

Example2

( ) ( )2 1-

x xf x and g x

x x

2 2 2 22( ( ))2 1( 2) 2 21

2

xx x x xxg f x g x

xx x x x xx

22 2 2 21( ( ))

21 2 2(1 ) 2 2 2 221

xx x x xxf g x f x

xx x x x xx

Homework

• pg 135: 13 – 31 odd

• Quiz: Tuesday, October 8

• Chapter 1 test: Friday, October 11