7.4 Composition of Functions
3/26/12
Function:
A function is a special relationship between values: Each
of its input values gives back exactly one output value.
It is often written as "f(x)" where x is the input value.
Ex 1: Given 𝑓 𝑥 = 3𝑥 + 1, find:
a.) 𝑓 0
Solution: Substitute 0 for x in 3x+ 1
3(0) + 1 = 1
d.) 𝑓 𝑎 + 3
𝑓 0 =1
3(a+3) + 1 = 3a + 9 +1
b.) 𝑓 ½
1
𝑓 𝑎 + 3 = 3a+10
3(½) + 1 = 1 + 1
2
e.) 𝑓 𝑎 + 3
1
𝑓 ½ =2
3𝑎 + 1 + 3
2
c.)
𝑓 𝑎
𝑓 𝑎 + 3 = 3a + 4
3(a) + 1
𝑓 𝑎 = 3a + 1
Ex 2: Given 𝑓 𝑥 = 𝑥 2 + 2𝑥 − 5, find:
a.) 𝑓 0
02 + 2(0) − 5,
𝑓 0 = -5
b.) 𝑓 −½
d.) 𝑓(3𝑥)
3𝑥 2 + 2 3𝑥 − 5
𝑓 3𝑥 = 9𝑥 2 + 6𝑥 − 5
1 2
1
−
+ 2 − − 5 e.) 3𝑓 𝑥 2
2
2
3(𝑥 + 2𝑥 − 5)
1
2 + 6𝑥 − 15
3𝑓
𝑥
=
3𝑥
−1−5
4
3
𝑓 −½ = -5
4
c.)
𝑓 𝑥+5
𝑥+5 2+2 𝑥+5 −5
𝑥 2 + 10𝑥 + 25 + 2𝑥 + 10 − 5
𝑓 𝑥 + 5 = 𝑥 2 + 12𝑥 + 30
y
8
g
6
f
4
2
x
–8
–6
–4
–2
2
4
6
8
–2
–4
Use the graph to find:
a. f(0)
Solution: at x = 0, the graph of the f (red) function is at 2.
f(0) = 2
b. g(0)
Solution: at x = 0, the graph of the g (black) function is at 3
g(0) = 3
Composition of functions
occurs when you insert one function into another. In effect, the range
(output) of the inside function becomes the domain (input) of the
outside function.
The notation for composition of functions is either
𝑓 𝑔 𝑥
Or ( 𝑓 ⃘ 𝑔) 𝑥
Note: 𝑓 𝑔 𝑥
≠ 𝑔(𝑓 𝑥 ) which means order matters!
y
8
g
6
f
4
2
x
–8
–6
–4
–2
2
4
6
8
–2
–4
Use the graph to find:
c.) f(g(0))
Solution: Evaluate g(0) first, and from the previous problem (b), g(0) = 3.
Then look at the graph of the f function and see what the y component
when x = 3.
At x = 3 the red graph is at -1
f(g(0)) = -1
y
8
g
6
f
4
2
x
–8
–6
–4
–2
2
4
6
8
–2
–4
Use the graph to find:
d.) g(f(0))
Solution: Evaluate f(0) which in problem (a) is 2. Then look at the graph
of the g function and see what the y component when x = 2.
At x = 2 the black graph is at 5.
g(f(0)) = 5
y
8
g
6
f
4
2
x
–8
–6
–4
–2
2
–2
–4
Use the graph to find:
a.)(f ⃘g)(-1) and (g ⃘f)(-1)
a.) Solution: Evaluate g(-1)
g(-1) = 2 then evaluate f(2)
f(2) = 0
b.) Solution: Evaluate f(-1)
f(-1) = 3 then evaluate g(3)
g(3) = 6
4
6
8
Given: 𝑓 𝑥 = 3 − 2𝑥 𝑎𝑛𝑑
𝑔 𝑥 = 𝑥 2 − 5𝑥 + 4
Evaluate the following expressions:
a. f(-1)
a.) 𝑓 −1 = 3 − 2 −1
b. g(-1)
=3+2=5
c. f(g(-1))
d. g(f(-1))
b.) 𝑔 −1 = (−1)2 −5 −1 + 4
= 1 + 5 + 4 = 10
c.) 𝑓 𝑔 −1 = 𝑓 10
= 3 − 2 10 = 3 − 20 = −17
d.) 𝑔 𝑓 −1 = 𝑔(5)
= (5)2 −5 5 + 4 = 25 − 25 + 4 = 4
Example 1
Add and Subtract Functions
Let f ( x ) = 4x 2 and g ( x ) = x + 1. Find:
a. f ( x ) + g ( x )
b. h ( x ) = f ( x ) – g ( x )
SOLUTION
a. h ( x ) = f ( x ) + g ( x )
b. h ( x ) = f ( x ) – g ( x )
= 4x 2 + ( x + 1)
= 4x 2 – ( x + 1)
= 4x 2 + x + 1
= 4x 2 – x – 1
In both parts ( a ) and ( b ), the domains of f and g are all
real numbers. So, the domain of h is all real numbers.
Example 2
Multiply and Divide Functions
Let f ( x ) = x 3 and g( x ) = 2x. Find:
f ( x)
b. h ( x ) =
g( x)
a. f ( x ) • g ( x )
𝑓 𝑥 ∙ 𝑔 𝑥 = 𝑥 3 ∙ 2𝑥
= 2x 4
𝑓 𝑥
𝑥3
=
𝑔 𝑥
2𝑥
1 2
= x
2
Checkpoint
Perform Function Operations
Let f ( x ) = 3x and g ( x ) = x – 1. Find
ANSWER
1. f ( x ) + g ( x )
2. f ( x ) • g ( x )
( x)
f
3.
g( x)
4x – 1
3x 2 – 3x,
3x
,x =1
x –1
Vocabulary
Composition of
Functions:
In Symbols:
Important
Note:
is the process of combining two functions
where one function is substituted in place
of each x in the other function.
f(g(x)) read as the “composition of f with g”.
g(f(x)) read as the “composition of g with f”.
f(g(x)) is not the same as g(f(x)).
The order of functions when they are
composed is very important.
Example 3
Write a Composition of Functions
Let f ( x ) = x 2 and g ( x ) = 2x + 3. Find the following.
a. f ( g ( x ))
b. g ( f ( x ))
SOLUTION
Write the composition by substituting the expression
for the inner function in the outer function, and simplify.
a. f ( g ( x )) = f ( 2x + 3) = ( 2x + 3)2 = 4x 2 + 12x + 9
b. g ( f ( x )) = g ( x 2 ) = 2( x 2 ) + 3 = 2x 2 + 3
Example 4
Evaluate a Composition of Functions
Let f ( x ) = x 2 + 3 and g ( x ) = 5x . Evaluate f ( g ( 2 )).
SOLUTION
To evaluate f ( g ( 2 )), first find g ( 2 ):
g ( 2 ) = 5( 2 ) = 10
Then substitute g ( 2 ) = 10 into f ( g ( 2 )):
f ( g ( 2 )) = f ( 10) = 102 + 3 = 100 + 3 = 103
Checkpoint
Find and Evaluate Compositions of Functions
Let f ( x ) = x 2 and g ( x ) = x – 1. Find the composition.
Then evaluate the composition when x = 2.
4. f ( g ( x ))
ANSWER
x 2 – 2x + 1; 1
5. g ( f ( x ))
ANSWER
x 2 – 1; 3
Example 5
Model a Real-World Situation
Hair Salon You have a coupon for
$10 off the cost of your purchase at
a hair salon. The salon also offers a
discount off your purchase, as
shown.
Let x be the cost of your purchase.
Then f ( x ) = x – 10 is the cost of the
purchase using your coupon, and
g ( x ) = 0.85x is the cost of your
purchase with the salon’s discount.
Find g ( f ( x )). Tell what it represents.
Example 5
Model a Real-World Situation
SOLUTION
g ( f ( x )) = g ( x – 10 ) = 0.85( x – 10 ) = 0.85x – 8.5
The composition g ( f ( x )) represents the cost of your
purchase when the $10 coupon is applied before the
15% discount.
Homework:
7.4 p.376 #14-24even, 26, 27
32-38even, 40-46 all