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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

Alg1A 03.10 Compound Functions

  • 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}

 

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