1.8
Combinations of Functions
JMerrill, 2010
Arithmetic Combinations
( f  g )( x)  f ( x)  g ( x)
Sum
Let f ( x)  5x  2 x  3
2
g ( x)  4 x  7 x  5
2
Find (f + g)(x)
(5 x  2 x  3)  (4 x  7 x  5)
2
9 x  5x  2
2
2
Difference
Let
( f  g )( x)  f ( x)  g ( x)
f ( x)  5 x 2  2 x  3
g ( x)  4 x 2  7 x  5
Find (f - g)(x)
(5 x  2 x  3)  (4 x  7 x  5)
2
x  9x  8
2
2
( fg )( x)  f ( x)  g ( x)
Product
Let
f ( x)  5 x
Find
( fg )( x)
g ( x)  3 x  1
2
5 x (3x  1)
2
15 x  5 x
3
2
 f 
f ( x)
where g ( x)  0
  ( x) 
g ( x)
g
Quotient
Let
Find
f ( x)  5 x
 f 
  ( x)
g
 5x 


 3x  1 
2
2
g ( x)  3 x  1
You Do: Let f ( x)  x  3, g ( x)  x  9
2
Find:
(f+g)(x)
x  x  12
2
(f-g)(x)
(f•g)(x)
x  3x  9 x  27
3
x  x  6
2
(g-f)(x)
x  x6
2
2
 f 
  ( x)
g 1
x3
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
Given f(x)  x and g(x)  4  x2
f
g
Find the domains of   (x) and   (x)
f
g


The domain of f(x) = [0, )
The domain of g(x) = [-2,2]
Example
f
f(x)
x

  x 
g(x)
4  x2
g


g(x)
g

  (x) 
f(x)
f
4  x2
x
Since the domains are: f(x) = [0, )
g(x) = [-2,2]
The domains of the quotients are
f
  : [0,2)
g
g
  : (0,2]
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
5
c(t )  ( f  32)
9
k (c)  c  273

This formula gives the Celsius
temp. that corresponds to the
Fahrenheit temp.
This formula converts the
Celsius temp. to Kelvins
This 2-step process that uses the output of
the first function as the input of the second
function.
Composition Notation

(f o g)(x) means f(g(x))

(g o f)(x) means g(f(x)
Composition of Functions:
A Graphing Approach (f g)(x) and (g
Find ( f g )( 1)
f ( g (1))
f (3)
f (3)  0
Find ( g f )(3)
g ( f (3))
g (0)
g (0)  2
f)(x)
You Do


f(g(0)) = 4
g(f(0)) = 4
f(x)

g(x)



(f°g)(3) = 3
(f°g)(-3) = 3
(g°f)(4) = 0
(f°g)(4) = 0
Compositions: Algebraically





Given f(x) = 3x2 and g(x) = 5x+1
Find f(g(2))
Find g(f(4))
How much is f(4)?
g(2)=5(2)+1 = 11
f(11) = 3(11)2
g(48) =
5(48)+1=241
=363
Compositions: Algebraically Con’t







Given f(x) = 3x2 and g(x) = 5x+1
Find f(g(x))
Find g(f(x))
What does f(x)=?
What does g(x)=?
f(5x+1)
g(3x2) =
5(3x2)+1=15x2+1
=3(5x+1)2
=3(25x2+10x+1)
=75x2+30x+3
You Do

f(x)=4x2-1

Find: (f g)(x)
g(x) = 3x
(g f)(x)
f(g(x)) 
g(f(x)) 
f(3x) 
g(4x2  1) 
4(3x)2  1 
4(9x )  1 
2
36x2  1
3(4x  1) 
2
12x2  3