Combinations of Functions & Inverse Functions

Obj: Be able to work with combinations/compositions of functions. Be able to find inverse functions.

TS: Making decisions after reflections and review

Warm-Up:Given f(x) = 2x + 3 and g(x) = x2 – 4, find…

f(2) + g(2)

g(3) – f(3)

Arithmetic Combinations

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

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

( )( ) ( ) ( )fg x f x g x

( )

( )

f f x

g g x

Where the domain is the real numbers that both f and g’s domains have in common.

For f/g also g(x) ≠ 0.

ExamplesFind each of the below combinations given

1) (f∙g)(2) 3) (f/g)(x) & its domain

2) (f – g)(2)

( )f x x2( ) 4g x x

Arithmetic Compositions (f○g)(x) = f(g(x))

Given the below find each of the following.

1) (f ○ g)(2) 2) g(f(-1))

1( )f x

x 2

1( )

2 3g x

x x

Arithmetic Compositions (f○g)(x) = f(g(x))

Given the below find each of the following.

3) f(g(x)) & its domain

1( )f x

x 2

1( )

2 3g x

x x

Find f(g(x)) and g(f(x))

( ) 2 3f x x 1( ) ( 3)

2g x x

Definition of the Inverse of a Function

Let f and g be two functions where 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. Under these conditions, g is the inverse of f and g is denoted f-1.

Thus f(f-1(x))=x and f-1(f(x))=x where the domain of f must equal the range of f-1 and the range of f must equal the domain of f-1.

Graphs of Inverses

Two equations are inverses if their graphs are reflections of one another across the line y=x.

x

y

1

1

y=xf

f -1

Inverse FUNCTIONSA function f(x) has an inverse function if the graph of f(x)

passes the ___________________.

(In other words the relation is ONE-TO-ONE: For each y there is exactly one x)

Circle the functions that are one-to-one (aka have inverse functions)

x

y

x

y

x

y

x

y

x

y

Examples:

Determine if the two functions f and g are inverses.

1) and 23

1)( xxg23)( xxf

Finding an Inverse

1)Verify that the function is one-to-one thus has an inverse function using the Horizontal Line Test.

2)Switch x & y.

3)Solve for y.

4)Make sure to use proper inverse notation for y for your final answer. (Ex: f-1(x), not y)

To check your answer:Verify they are inverses by testing to see if f(f-1(x)) = f-1(f(x)) = x

Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the

inverse function .

1) f(x) = -.5x + 3

Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the

inverse function .

2) f(x) = x2 – 4