Combinations of Functions
Sec. 1.8
Objective
You will learn how to find arithmetic combinations and
compostiions of functions.
Arithmetic Combinations of Fns
Two functions can be combined by the operations of
addition, subtraction, multiplication, and division to
create new functions.
The domain of an arithmetic combination of functions
f and g consists of all real numbers that are common
to the domains of f and g.
In the case of the quotient, there is the further
restriction that g(x) ¹ 0 .
Arithmetic Combinations of Fns
Sum:
( f + g)(x) = f (x)+ g(x)
Difference:
Product:
( f - g)(x) = f (x) - g(x)
( fg)(x) = f (x)·g(x)
æfö
Quotient: ç ÷ (x) = f (x)
g(x)
ègø
g(x) ¹ 0
Example
Let f (x) = 7x - 5 and
Find (f – g)(4).
g(x) = 3- 2x .
( f - g)(4) = f (4)- g(4)
= (7x - 5) - (3- 2x)
= 7x - 5-3+ 2x
= 9x -8
Example
Let f (x) = x 2 and g(x) = 2 - x .
Find (f + g)(x). Then evaluate the sum when x = 3.
( f + g)(x) = f (x)+ g(x)
2
= (x ) + (2 - x)
2
= x - x+2
2
( f + g)(3) = (3) - (3)+ 2
= 9 - 3+ 2
=8
Example
Let f (x) = x 2 and g(x) = 2 - x .
Find (f – g)(x). Then evaluate the sum when x = 2.
( f - g)(x) = f (x) - g(x)
2
= (x ) - (2 - x)
2
= x + x-2
2
( f - g)(2) = (2) + (2) - 2
= 4+2-2
=4
Example
Let f (x) = 5x + 2 and g(x) = 3x - 4 .
Find (fg)(x). Then evaluate the product when x = 2.
( fg)(x) = f (x)·g(x)
= (5x + 2)(3x - 4)
2
=15x - 20x + 6x -8
2
=15x -14x -8
2
( fg)(2) =15(2) -14(2) -8
= 60 - 28-8
= 24
Example
Let f (x) = x and g(x) =1- x .
Find (f/g)(x). Then find the domain.
2
æfö
f (x)
ç ÷ (x) =
g(x)
ègø
x2
=
1- x
g(x) ¹ 0
x ¹1
Domain
1- x = 0
-x = -1
x =1
Practice Problems
Sec. 1.8, page 81 – 83
# 5 – 11 odd, 13, 17, 21, 24
Composition of Functions
The composition of the function f with the function
g is defined as
The domain of
is the set of all x in the
domain of g such that g(x) is in the domain of f.
Example
Let f(x)= 3x + 4 and g(x)= 2x2 – 1. Find
a.)
= f ( g ( x ))
2
= f ( 2x -1)
= 3( 2x 2 -1) + 4
2
= 6x - 3+ 4
2
= 6x +1
Example
Let f(x)= 3x + 4 and g(x)= 2x2 – 1. Find
b.)
= g ( f ( x ))
= g (3x + 4)
2
= 2 (3x + 4) -1
2
= 2(9x + 24x +16) -1
2
=18x + 48x + 32 -1
2
=18x + 48x + 31
Example
Let f(x)= 3x + 4 and g(x)= 2x2 + 2. Find
a.)
= f ( g ( x ))
2
= f ( 2x + 2)
= 3( 2x 2 + 2) + 4
2
= 6x + 6 + 4
2
= 6x +10
Example
Let f(x)= 3x + 4 and g(x)= 2x2 + 2. Find
b.)
= g ( f ( x ))
= g (3x + 4)
2
= 2 (3x + 4) + 2
2
= 2(9x + 24x +16)+ 2
2
=18x + 48x + 32 + 2
2
=18x + 48x + 34
Example
Let f(x)= 3x + 4 and g(x)= 2x2 + 2. Find
c.)
= f ( g (-2))
2
= f ( 2(-2) + 2)
= f (10)
= 3(10) + 4
= 34
Example
Find the composition
f (x) = x
. Then find the domain of
g(x) = x - 5
= f ( g ( x ))
= f ( x - 5)
= x-5
Domain
x-5³ 0
x³5
Example
Find two functions f and g such that
4- x
h(x) =
9
.
Application
The function f(x)= 0.06x represents the sales tax owed on a
purchase with a price tag of x dollars and the function g(x)= 0.75x
represents the sale price of an item with a price tag of x dollars
during a 25% off sale. Using one of the combinations of functions,
write the function that represents the sales tax owed on an item
with a price tag of x dollars during a 25% off sale.
= f ( g ( x ))
= f ( 0.75x )
= 0.06 ( 0.75x )
Practice Problems
Sec. 1.8, page 81 – 83
# 5 – 11 odd, 13, 17, 21, 24
# 31, 39, 43, 52