NEW FUNCTIONS FROM OLDNEW FUNCTIONS FROM OLD

1

1.3New Functions from Old

FunctionsIn this section, we will learn:

How to obtain new functions from old functions

and how to combine pairs of functions.

FUNCTIONS AND MODELS

Two functions f and g can be combined

to form new functions f + g, f - g, fg, and

in a manner similar to the way we add,

subtract, multiply, and divide real numbers.

COMBINATIONS OF FUNCTIONS

f

g

The sum and difference functions are defined by:

(f + g)x = f(x) + g(x) (f – g)x = f(x) – g(x)

If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection .

This is because both f(x) and g(x) have to be defined.

A B

SUM AND DIFFERENCE

For example, the domain of

is and the domain of

is .

So, the domain of

is .

( )f x x[0, )A ( ) 2g x x

( , 2]B

( ) 2f g x x x [0,2]A B

SUM AND DIFFERENCE

Similarly, the product and quotient

functions are defined by:

The domain of fg is . However, we can’t divide by 0. So, the domain of f/g is

( )( )( ) ( ) ( ) ( )

( )

f f xfg x f x g x x

g g x

A B

| ( ) 0 .x A B g x

PRODUCT AND QUOTIENT

For instance, if f(x) = x2 and g(x) = x - 1,

then the domain of the rational function

is ,

or

2( / )( ) /( 1)f g x x x | 1x x

( ,1) (1, ).

PRODUCT AND QUOTIENT

There is another way of combining two

functions to obtain a new function.

For example, suppose that and

Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.

We compute this by substitution:

( )y f u u 2( ) 1. u g x x

COMBINATIONS

2 2( ) ( ( )) ( 1) 1y f u f g x f x x

This procedure is called composition—

because the new function is composed

of the two given functions f and g.

COMBINATIONS

In general, given any two functions f and g,

we start with a number x in the domain of g

and find its image g(x).

If this number g(x) is in the domain of f, then we can calculate the value of f(g(x)).

The result is a new function h(x) = f(g(x)) obtained by substituting g into f.

It is called the composition (or composite) of f and g. It is denoted by (“f circle g”).

COMPOSITION

f g

Given two functions f and g,

the composite function

(also called the composition of f and g)

is defined by:

f g

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

DefinitionCOMPOSITION

The domain of is the set of all x

in the domain of g such that g(x) is in

the domain of f. In other words, is defined whenever

both g(x) and f(g(x)) are defined.( )( )f g x

f gCOMPOSITION

The figure shows

how to picture

in terms of machines.

COMPOSITION

f g

If f(x) = x2 and g(x) = x - 3, find

the composite functions and .

We have:

f g g f

2( )( ) ( ( )) ( 3) ( 3)f g x f g x f x x

Example 6COMPOSITION

2 2( )( ) ( ( )) ( ) 3g f x g f x g x x

You can see from Example 6 that,

in general, .

Remember, the notation means that, first,the function g is applied and, then, f is applied.

In Example 6, is the function that first subtracts 3 and then squares; is the function that first squares and then subtracts 3.

f g g f

COMPOSITION Note

f g

g ff g

If and ,

find each function and its domain.

a.

b.

c.

d.

( )f x x ( ) 2g x x

f g

Example 7COMPOSITION

g f

f fg g

The domain of is:

4( )( ) ( ( )) ( 2 ) 2 2f g x f g x f x x x

f g

| 2 0 | 2 ( , 2]x x x x

COMPOSITION Example 7 a

For to be defined, we must have . For to be defined, we must have ,

that is, , or . Thus, we have . So, the domain of is the closed interval [0, 4].

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

x 0x 2 x 2 0x

2x 4x 0 4x

g f

Example 7 bCOMPOSITION

The domain of is .

COMPOSITION Example 7 c

4( )( ) ( ( )) ( )f f x f f x f x x x

f f [0, )

This expression is defined when both and .

The first inequality means . The second is equivalent to , or ,

or . Thus, , so the domain of is

the closed interval [-2, 2].

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

2 0x 2 2 0x

2x 2 2x 2 4x

2x 2 2x g g

Example 7 dCOMPOSITION

It is possible to take the composition

of three or more functions.

For instance, the composite function is found by first applying h, then g, and then f as follows:

f g h

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

COMPOSITION

Find if ,

and .

f g h ( ) /( 1)f x x x 10( )g x x ( ) 3h x x

( f og oh)(x) f (g(h(x)))

f (g(x 3)) f ((x 3)10 ) (x 3)10

(x 3)10 1

Example 8COMPOSITION

So far, we have used composition to build

complicated functions from simpler ones.

However, in calculus, it is often useful

to be able to decompose a complicated

function into simpler ones—as in the

following example.

COMPOSITION

Given , find functions

f, g, and h such that .

Since F(x) = [cos(x + 9)]2, the formula for F states: First add 9, then take the cosine of the result, and finally square.

So, we let:

2( ) cos ( 9)F x x F f g h

2

( ) 9

( ) cos

( )

h x x

g x x

f x x

Example 9COMPOSITION

Then,

2

( )( )

( ( ( )))

( ( 9))

(cos( 9))

[cos( 9)]

( )

f g h x

f g h x

f g x

f x

x

F x

COMPOSITION Example 9