Algebra 2H
Unit 6
Name___________________________________________
6.6 Notes: Operations with Functions
Essential Understanding: In this lesson, you will be learning how to add, subtract, multiply, and
divide two functions to create new functions. The operations with functions work very similarly to
operations with real numbers. One key difference, however, is that you must consider the domain
of each function you are working with when you determine the domain of the new function you are
creating.
Example 1: Adding and Subtracting Functions.
Let 𝑓(𝑥) = 5𝑥 3 + 1 and 𝑔(𝑥) = √𝑥 − 4. What are f + g and f – g? What are their domains?
a. f  g  ________________
Domain:_________________
b. f  g  ________________
Domain:_________________
Example 2: Multiplying and Dividing Functions
Let 𝑓(𝑥) = 𝑥 2 + 𝑥 − 6 and g ( x ) 
1
𝑓
. What are f  g and ? What are their domains?
𝑔
x2
a. f  g  ________________
Domain:_________________
f
 ________________
g
Domain:_________________
b.
1. Let f ( x)  3x 2  11x  4 and g ( x)  3x  1 . Find a) f  g , b) f  g , c) f  g , and d)
f
as well
g
as each of their domains.
a)
b)
c)
d)
2. Refer to problem #1 above. Is the domain of
g
f
different than the domain of
? If so, explain.
f
g
3. Let f (x) = 3x2 and g(x) = 2 – 5x. Perform each function operation. State the domain of the resulting
function.
a) 2 f ( x)  3 g ( x)
b) f ( x)  2 g ( x)
c) 3 f ( x)  4
d) 2 f ( x)  g ( x)
e)
2 f ( x)
g ( x)
f)
2 f ( x)  f ( x )
g ( x)
4. Use the graphs of f(x) and g(x) below to evaluate the following:
a) f (4)  ________
b) g (8)  ________
c) ( f  g )(2)  ________
d) ( f  g )(2)  ________
e) ( f  g )(2)  ________
f 
 (2)  ________
g
f) 
6.6 The Mechanics of Composite Functions
ESSENTIAL UNDERSTANDING: When the output from one function becomes the input for
another function, you have composed the two functions. Composite functions help us to simplify
computations when multiple functions are used.
Objective: To find the composite of two functions.
Example 1: Evaluate a Composite Function
Let f (x) = x2 + 1 and g (x) = x – 2. What is g o f 2?
Let f (x) = x2 and g(x) = 3x + 1. Evaluate each expression.
a. ( f g )(0)
b. ( f g )(2)
c. ( g f )(2)
d. ( g f )(1)
e. ( f f )(3)
f. (g g)(4)
g. Is this composition commutative? Explain using examples from above.
Example 2: Finding Composite Functions
Let f (x) = x2 + 2x + 1 and g (x) = x – 2. What is g o f x  and  f og x ?
Got it? Let f (x) = x2 and g(x) = 3x + 1. Evaluate each expression.
a. ( f g )( x )
b. ( g f )( x )
c. ( f g)(a  2)
d. (g f )(x  4)
Suppose f(x) = x2 + 3x + 4. Find f(g(x)) if g(x) is the following:
a. g(x) = 3
b. g(x) = y
c. g(x) = ant
d. g(x) = √x
e. g(x) = x+2
Example 3: Using Composite Functions
4
3
The formula V   r 3 expresses the relationship between the volume V and radius r of a sphere. A
weather balloon is being inflated so that the radius is changing with respect to time according to the
equation r  t 1 , where t is the time, in minutes, and r is the radius, in feet.
a. Write a composite function f (t) to represent the volume of the weather balloon after t
minutes. Do not expand the expression.
b. Find the volume of the balloon after 5 minutes. Round the answer to two decimal places.
Got It? Suppose Lili’u works at Nordstrom’s in Bell Square. Three times per year she is
allowed to combine her employee discount with special sale prices. Let x be the retail price of
a blouse.
a.
Lili’u’s employee discount is 20%. Write a function E(x) that represents the cost of the blouse
after the discount.
b.
Due to a manufacturer’s incentive, the blouse is marked down 25%. Write a function M(x)
that represents the sale price.
c.
The sales tax on clothing is 6%. Write a function T (x) that describes the cost of a clothing item
with sales tax included.
d.
Lili’u found an $80 blouse to which the discounts apply. Use the function composition
f x  T E M x  to write the function f (x) that represents the price Lili’u will pay for
the blouse. How much will she pay?
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