Title: Composite Functions Topic: Functions Difficulty: Required Skills: Basic math operations, defining and calling functions, Language: Python

In this activity students will write their own composite functions. They will also write a few composite functions based on real-world type situations. The answers are included below, but students would be given only the comments as a guide.

 # Create two functions f and gdef f(x):  return(x**2+2)  def g(x):  return(3*x-1)# Create the composite function f(g(x)) and call it composite_fg. You can solve for f(g(x)) on paper.def composite_fg(x):  return(9*x**2-6*x+3)# Compute f(g(3)) using both functions f and g separately and by using composite_fg. Are they equal?print(f(g(3)))print(composite_fg(3))# Create the composite function g(f(x)) and call it composite_gf.def composite_gf(x):  return(3*x**2+5)# Compute g(f(5)) using both functions g and f separately and by using composite_gf. Are they equal?print(g(f(5)))print(composite_gf(5))# Examples # You work forty hours a week at a furniture store. You receive a \$220 weekly salary, plus a 3% commision on sales over \$5000. Assume that you sell enough this week to get the commission. Use composite functions to calculate how much you take home in commission when you sell \$8000.def amount_over(sales):  return sales - 5000  def commision(amt):  return amt*0.03print(commision(amount_over(8000)))# You make a purchase at a local hardware store, but what you've bought is too big to take home in your car. For a small fee, you arrange to have the hardware store deliver your purchase for you. You pay for your purchase, plus the sales taxes, plus the fee. The taxes are 7.5% and the fee is \$20.  # Part a - Write a function t(x) for the total, after taxes, on the purchase amount x. Write another function d(x) for the total, including the delivery fee, on the purchase amount x.def t(x):  return(x + 0.075*x)  def d(x):  return(x+20)  # Part b - Calculate and interpret d(t(x)) and t(d(x)). Which results in a lower cost to you?print(d(t(20)))print(t(d(20)))# ANSWER d(t(x)) results in a lower cost because the tax is not being applied to the delivery fee# Part c - Suppose taxes, by law, are not to be charged on delivery fees. Which composite function must then be used?# ANSWER d(t(x) must be used