Composite and Inverse Functions

3rd

4th Block

Composite Functions

• In a composition of functions, the range of one function is part of the domain of the other function

• Basically substituting one function into another function

• Notation of composite functions is

• f(g(x)) or (f◦g)(x)

• Read as “f of g of x”

Composite Functions

)2)(())((

))(())((

...

4)(

12)(34)( 2

fhgxhg

xfgxgf

find

xxh

xxgandxxxf

Composite Functions

)1)(())((

))(())((

...

)(

13)(2)( 2

fhxhg

xfgxgf

find

xxh

xxgandxxxf

Inverse Functions

• The domain of the inverse is the range of the original function

• The range of the inverse is the domain of the original function

• Switching x and y

To determine whether the inverse is a function…

• Switch x and y values and determine whether the domain of inverse is paired with only one value in the range (domain can not repeat)

:Inverse

(3,3) (-3,3) (5,2) (4,2) (2,3) :Relation

function a isit whether decide andrelation theof inverse theDetermine

Determining whether the inverse is a function from the graph of the original

• You do the vertical line test to determine whether the graph represents a function

• To determine whether its inverse would be a function, then you would do a horizontal line test

Finding the Inverse of a Relation

What is the inverse of relation s?

Relation s

X Y

0 -1

2 0

3 2

4 3

To find the inverse of a function

• Notation: f -1(x)=

• Steps:

1. Replace f(x) with y

2. Switch the x and y variable

3. Solve for y

4. Replace y with f -1(x)=

Find the following

)2(

)()(

)()(

3

2)(1)(

25)(32)(

1

11

11

2

f

xmxh

xgxf

xxmxxh

xxgxxf

Find the following

)3(

)()(

)()(

5)(2)(

1)(14)(

1

11

11

2

f

xmxh

xgxf

xxmxxh

xxgxxf

Inverse functions

?

25

1)(105)(

))(())((

inversestheyare

xxgxxf

inversesarefunctionstwothethen

xxfgandxxgfif

Are They Inverses?

• 𝑓 𝑥 = 4𝑥 + 6

• 𝑔 𝑥 =𝑥−6

4

Are They Inverses?

• 𝑓 𝑥 = −6𝑥

• 𝑔 𝑥 =1

6𝑥