1.41.41.41.4

Combinations of FunctionsCombinations of Functions

Quick Review

Find the domain of the function and express it in interval notation.

11. ( )

4

2. ( ) 1

13. ( )

1

4. ( ) log

5. ( ) 4

xf x

x

f x x

f xx

f x x

f x

What you’ll learn about• Combining Functions Algebraically• Composition of Functions• Relations and Implicitly Defined Functions

… and whyMost of the functions that you will encounter incalculus and in real life can be created by

combining ormodifying other functions.

The Identity Function

The Squaring Function

The Cubing Function

The Reciprocal Function

The Square Root Function

The Exponential Function

The Natural Logarithm Function

The Sine Function

The Cosine Function

The Absolute Value Function

The Greatest Integer Function

The Logistic Function

Example Looking for Domains

One of the functions has domain the set of all reals except 0.

Which function is it?

Example Looking for Domains

One of the functions has domain the set of all reals except 0.

Which function is it?

The function 1/ has a vertical asymptote at 0.y x x

Example Analyzing a Function Graphically

2Graph the function ( -3) . Then answer the following questions.

(a) On what interval is the function increasing?

(b) Is the function even, odd, or neither?

(c) Does the function have any extrema?

y x

Example Analyzing a Function Graphically

2Graph the function ( -3) . Then answer the following questions.

(a) On what interval is the function increasing?

(b) Is the function even, odd, or neither?

(c) Does the function have any extrema?

y x

(a) The function is increasing on [3, ).

(b) The function is neither even or odd.

(c) The function has a minimum value of 0 at 3.x

Sum, Difference, Product, and Quotient

Let and be two functions with intersecting domains. Then for all values

of in the intersection, the algebraic combinations of and are defined

by the following rules:

Sum: ( ) ( )

Differ

f g

x f g

f g x f x g x

ence: ( ) ( ) ( )

Product: ( )( ) ( ) ( )

( )Quotient: , provided ( ) 0

( )

In each case, the domain of the new function consists of all numbers that

belong to both the domain of and

f g x f x g x

fg x f x g x

f f xx g x

g g x

f

the domain of . g

Example Defining New Functions

Algebraically3Let ( ) and ( ) 1. Find formulas of the functions

(a)

(b)

(c)

(d) /

f x x g x x

f g

f g

fg

f g

Example Defining New Functions Algebraically

3Let ( ) and ( ) 1. Find formulas of the functions

(a)

(b)

(c)

(d) /

f x x g x x

f g

f g

fg

f g

3

3

3

3

(a) ( ) ( ) 1 with domain [ 1, )

(b) ( ) ( ) 1 with domain [ 1, )

(c) ( ) ( ) 1 with domain [ 1, )

( )(d) with domain ( 1, )

( ) 1

f x g x x x

f x g x x x

f x g x x x

f x x

g x x

Composition of Functions

Let and be two functions such that the domain of intersects the range

of . The composition of , denoted , is defined by the rule

( )( ) ( ( )).

The domain of consists of all -values

f g f

g f g f g

f g x f g x

f g x

in the domain of that

map to ( )-values in the domain of .

g

g x f

Composition of Functions

Example Composing Functions

Let ( ) 2 and ( ) 1. Find

(a)

(b)

xf x g x x

f g x

g f x

Example Composing Functions

Let ( ) 2 and ( ) 1. Find

(a)

(b)

xf x g x x

f g x

g f x

1(a) ( ( )) 2

(b) ( ( )) 2 1

x

x

f g x f g x

g f x g f x

Example Decomposing Functions

2

Find and such that ( ) ( ( )).

( ) 5

f g h x f g x

h x x

Example Decomposing

Functions 2

Find and such that ( ) ( ( )).

( ) 5

f g h x f g x

h x x

2

2

One possible decomposition:

( ) and ( ) 5

Another possibility:

( ) 5 and ( )

f x x g x x

f x x g x x

Example Using Implicitly Defined

Functions2 2Describe the graph of the relation 2 4.x xy y

Example Using Implicitly Defined

Functions

2 2Describe the graph of the relation 2 4.x xy y

2 2

2

2 4

( ) 4 factor the left side

2 take the square root of both sides

2 solve for

The graph consists of two lines 2 and 2.

x xy y

x y

x y

y x y

y x y x