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4.7 – Arithmetic of Functions
Objectives
 Add, subtract, multiply, and divide functions.
 Combine two or more functions to form a new function.
 Apply the arithmetic of functions to solve problems.
Key Terms
 Commutative Property – A rule stating that adding or multiplying two numbers in either order will not change the answer.
 The commutative property of addition can be written as a + b = b + a.
 Ex: 4 + 3 = 3 + 4, because 7 = 7
 The commutative property of multiplication can be written as a * b = b * a.
 Ex: 4 * 3 = 3 * 4, because 12 = 12
 Composite function – A function that is created by using the output of one function as the input of another.
 A composite function is a more complicated function created from combining simpler functions.
Notes
Transformations of Functions 
Addition
Subtraction
Multiplication
Division
 Example: Leadership is planning an event.
 Their goal is to raise enough funds to buy new tshirts.
 The ticket price for the event is $15; however, the venue rental is $350 and each chair costs $5 to rent.
 How much profit will Leadership make on this event to buy new tshirts?
 Ticket Sales:
 Rental Costs:
 x = person attending event
 Income $ (ticket sales) – Outgoing $ (rentals)
 Setup:
 Distribute the negative sign:
 Simplify:
 Profit for each ticket sale is: P(x) = 10x – 350
 Test
 What if they only sell 20 tickets?: P(20) = 10(20) – 350 = – 150
 They will lose $150 dollars!
 What if they sell 50 tickets?: P(50) = 10(50) – 350 = 150
 They will make $150 dollars!
 How many tickets will they need to sell to break even?
 Setup: 10x – 350 = 0 (add 350 to both sides)
 10x = 350 (divide by 10 on both sides)
 x = 35 tickets to break even!
 Rules
 To compose two functions, substitute one function for the variable into the other functions. Then, simplify!
More Examples
 Example 1
 Given
 Find: (fg)(x): =
 Answer:
 Example 2
 Given
 Find (f of g)(x):
 Answer:
 Example 3
 A student gets paid to sell food and drinks at the school play during intermission. She earns an hourly rate of $12, plus an extra $0.50 for each bakery item she sells and $0.25 for each drink she sells. If h = hours, b = bakery items, and d = drinks, what function can she use to calculate her earnings?
 Note the Facts
 $12 per hour
 $0.50 each bakery item
 $0.25 each drink
 h: hours
 b: bakery items
 d: drinks
 Q?: What function can be used to calculate how much she earns (T: total earnings)?
 Setup the Problem

 Remember: 50 cents equals half of a dollar (50 out of 100 cents), so you can write it as which reduces to
 Also 25 cents equals one quarter of a dollar (25 out of 100 cents), so you can write it as which reduces to
 When you multiply a fraction by a variable, multiply the numerators together to combine them into one numerator, then multiply the denominators together to combine them into one denominator. If there is no denominator, assume the denominator is 1.
 and
 Answer:
 Example 4
 Find: G(F(x)):
 Answer:
 You can also FOIL it:
 =
 =

Important!
Practice (Apex Study 4.7)
 Try practice problems on Pg 8, 13, 20
 Mandatory: write and answer problems on Pg 14, 21
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