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Another exciting episode in your continuing Pre-Calculus experience!

1.8 Combinations of Functions:Composite Functions

Homework for section 1.8

P88 13-27, 37-59, 73

Two functions may behave exactly like two numbersin that they may be added, subtracted, multiplied, and divided.

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( ) ( ) ( )

( )( )

( )

f g x f x g x

f g x f x g x

fg x f x g x

ff xx

g g x

Lets say the we have the following functions:

f(x) = x+5 g(x) = 3x

f(x) + g(x) = ? (f+g)(x) = 4x+5

f(x) g(x) = ? (f g)(x) = 3x2+15x

(f g)(-5) = ?

( 5 ) ( 5 ) 5 and ( 5 ) 3( 5 )

( 5 ) and ( 5 ) 0 1 ( )( 5 )5 0

f g

f g f g

or a parenthesis? But, do we use a bracket

The farthest right we can even think of going on the number line is:The farthest left we can even think of going on the number line is:02

To find the quotients’ domains, we first haveto find the domain of f(x) and g(x) individually.The domain of the quotient is the intersection of the domain of f(x) and g(x).

f x

xg x

24

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

g x

xxf

24

Find the domain of each of these combinations…

D: D:

or a parenthesis? But, do we use a bracket

D of :f

xg

D of :g

xf[0, 2) (0 , 2]

[ 0, ) 2, 2

It’s important to remember:

ANY restriction on functions f and g

MUST be considered when forming the:

Sum, Difference, Product, or Quotient of f and g.

Composition of Functions : just another way to combine functions…

If f(x) = x2 and g(x) = x + 1 , then the

composition of f with g is:

(f o g)(x) - pronounced “the f of the g of x”. (Or as I like to call it: fog x.)

(f o g)(x) = f(g(x)) = f(x + 1) = (x + 1)2

* Domain of f o g is all x values in the domainof g where g(x) is in the domain of f.

what the heck does that mean?????

Do this by working from the inside out…

2 2Let 9 and ( ) 9f x x g x x

a) find (f o g)(x)b) find the domain of (f o g)(x)

29f x a) ( )f g x 2

29 9x 2x

b)

or, in interval notation:

-3 x 3

3, 3

Why? The domain of g(x) is [-3,3]. Theseare the only values you can even think of trying to fit into the domain of (f o g)(x).

1

Let and ( )2

f x g x xx

a) find (f o g)(x)b) find the domain of (f o g)(x)

f x a) ( )f g x

12x

b) all non-negative numbers

( ) domaing x added to fit

the (f g)( ) domainx

ex cept for 4.x

D: [ 0, 4) , (4, )

Identifying a composite function: it is used in calculus - need to be able to determine which two functions make up the composite function.You must be the function…………

2Ex press as a composition

of two other functio

1h

2

ns.

xx

This means we must find: f(x) and g(x) such that their composition gives us h(x)…

so that: (f o g)(x) =h(x)

the g("Find" the inner function first ( ), and then the

ou

x )

the ftter ( (x ) ).

Lets let g x

Lets let f x

2

1h

2x

x

Go! Do!