# 3.10 – 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 does NOT work for subtraction or division.
• Commutative Property of Addition: a + b = b + a
• Ex. 4 + 7 = 7 + 4 b/c 11 = 11
• Commutative Property of Multiplication: ab = ba
• Ex.  4 • 7 = 7 • 4 b/c 28 = 28

Notes

• Example 1: Combined Functions with Addition
• If $f(x)=5x^3$ and $g(x)=2x+1$, find $(f+g)(x)$
• Ans: $5x^3+2x-1$

• Example 2: Combined Functions with Multiplication
• If $f(x)=5x^3$ and $g(x)=x+1$, find $(f \cdot g)(x)$
• Ans: $5x^4+5x^3$

• Example 3: Combined Functions with Subtraction
• If $f(x)=4x+1$ and $g(x)=x^2-5$, find $(f-g)(x)$
• Ans: $-x^2+4x+6$

• Example 4: Combined Functions with Distribution (Multiplication)
• If $f(x)=4x+1$ and $g(x)=x^2-5$, find $(f \cdot g)(x)$
• Rule: distribute each term in the 1st binomial one at a time.
• Ans: $4x^3+x^2-20x-5$

• Example 5: Combined Functions with Division
• If $f(x)=4x+1$ and $g(x)=x^2-5$, find $(\frac{f}{g})(x)$
• Remember: the denominator can never be zero!
• Remember: square roots can never be negative
• Ans: $\frac{4x+1}{x^2-5}, x \neq \pm \sqrt{5}$

• Example 6: Creating a Combined Function from a Real World Scenario
• 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, write a function that she can use to calculate her earnings? • Facts • 12 per hour is 12h. •$0.50 for 1 bakery item. 0.50 is the same as 50/100 (reduced to 1/2), so b/2.
• \$0.25 for 1 drink item. 0.25 is the same as 25/100 (reduced to 1/4), so d/4.
• Ans: $T=12h+ \frac{b}{2}+ \frac{d}{4}$