1.8 Combinations of
Functions
JMerrill, 2010
Arithmetic Combinations
SumLet
Find (f + g)(x)
( )( ) ( ) ( )f g x f x g x 2 2( ) 5 2 3 ( ) 4 7 5 f x x x g x x x
2 2
2
(5 2 3) (4 7 5)
9 5 2
x x x x
x x
DifferenceLet
Find (f - g)(x)
( )( ) ( ) ( )f g x f x g x 2 2( ) 5 2 3 ( ) 4 7 5 f x x x g x x x
2 2
2
(5 2 3) (4 7 5)
9 8
x x x x
x x
ProductLet
Find
( )( ) ( ) ( ) fg x f x g x2( ) 5 ( ) 3 1 f x x g x x
( )( )fg x
2
3 2
5 (3 1)
15 5
x x
x x
Quotient Let
Find
( )( ) ( ) 0( )
f f xx where g xg g x
2( ) 5 ( ) 3 1 f x x g x x
( )f xg
253 1xx
You Do: Let Find:
(f+g)(x)(f•g)(x)
(f-g)(x)
(g-f)(x)
2( ) 3, ( ) 9 f x x g x x
( )f xg
2 12x x
2 6x x
2 6x x
3 23 9 27x x x
13x
Finding the Domain of Quotients of Functions To find the domain of the quotient,
first you must find the domain of each function. The domain of the quotient is the overlap of the domains.
Example
The domain of f(x) = The domain of g(x) = [-2,2]
2Given f(x) x and g(x) 4 xgfFind the domains of (x) and (x)g f
[0, )
Example
Since the domains are: f(x) = g(x) = [-2,2] The domains of the quotients are
2ff (x) xxg g(x) 4 x
[0, )
2g g(x) 4 x(x)ff (x) x
gf :[0,2) : (0,2]g f
Composition of Functions Most situations are not modeled by
simple linear equations. Some are based on a system of functions, others are based on a composition of functions.
A composition of functions is when the output of one function depends on the input from another function.
Compositions Con’t For example, the amount you pay on
your income tax depends on the amount of adjusted gross income (on your Form 1040), which, in turn, depends on your annual earnings.
Composition Example In chemistry, the process to convert
Fahrenheit temperatures to Kelvin units
This 2-step process that uses the output of the first function as the input of the second function.
5( ) ( 32)9
c t f This formula gives the Celsius temp. that corresponds to the Fahrenheit temp.
( ) 273k c c This formula converts the Celsius temp. to Kelvins
Composition Notation (f o g)(x) means f(g(x))
(g o f)(x) means g(f(x)
Composition of Functions: A Graphing Approach
( )( 1)Find f g ( ( 1))f g (3)f(3) 0f
( )(3)Find g f( (3))g f(0)g(0) 2g
(f g)(x) and (g f )(x)
You Do f(g(0)) = g(f(0)) =
(f°g)(3) = (f°g)(-3) =
(g°f)(4) = (f°g)(4) =
f(x)
g(x)
44
33
00
Compositions: Algebraically Given f(x) = 3x2 and g(x) = 5x+1 Find f(g(2)) Find g(f(4)) g(2)=5(2)+1 = 11 f(11) = 3(11)2
=363
How much is f(4)?g(48) = 5(48)+1=241
Compositions: Algebraically Con’t Given f(x) = 3x2 and g(x) = 5x+1 Find f(g(x)) Find g(f(x)) What does g(x)=? f(5x+1) =3(5x+1)2
=3(25x2+10x+1) =75x2+30x+3
What does f(x)=?g(3x2) = 5(3x2)+1=15x2+1
You Do f(x)=4x2-1 g(x) = 3x
Find: (f g)(x) (g f )(x)
2
2
2
f (g(x))f(3x)4(3x) 14(9x ) 136x 1
2
2
2
g(f(x))g(4x 1)
12x 33(4x 1)