# 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
• Combining Functions

Subtraction

Multiplication

Division

• Example: Leadership is planning an event.
• Their goal is to raise enough funds to buy new t-shirts.
• 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 t-shirts? • Ticket Sales: $f(x)=15x$ • Rental Costs: $g(x)=5x+350$ • x = person attending event • Income$ (ticket sales) – Outgoing $(rentals) • Setup: $(f-g)(x)=15x-(5x+350)$ • Distribute the negative sign: $(f-g)(x)=15x-5x-350$ • Simplify: $10x-350$ • 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 • $f(x)=2x+1$ • $g(x)=x^2-7$ • Find: (f-g)(x): $2x+1-(x^2-7)$ = $2x+1-x^2+7$ • Answer: $-x^2+2x+8$ • Example 2 • Given • $f(x)=8x$ • $g(x)=2x+1$ • Find (f of g)(x): $f(g(x))=8(2x+1)$ • Answer: $16x+8$ • 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
• $T=12h+0.50b+0.25d$
• Remember: 50 cents equals half of a dollar (50 out of 100 cents), so you can write it as $\frac{50}{100}$ which reduces to $\frac{1}{2}$
• Also 25 cents equals one quarter of a dollar (25 out of 100 cents), so you can write it as $\frac{25}{100}$ which reduces to $\frac{1}{4}$
• 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.
• $\frac{1}{2}\cdot\frac{b}{1}=\frac{b}{2}$ and $\frac{1}{4}\cdot\frac{d}{1}=\frac{d}{4}$
• Answer: $T=12h+\frac{b}{2}+\frac{d}{4}$

• Example 4
• $F(x)=x-5$
• $G(x)=x^2$
• Find: G(F(x)): $G(x-5)$
• Answer: $(x-5)^2$
• You can also FOIL it: $(x-5)(x-5)$
• = $x^2-5x-5x+25$
• = $x^2-10x+25$