7.1: Combining Functions
Contents
Objectives:[edit]
- Find formulas for combinations of functions
Important Items[edit]
Definitions:[edit]
Notes:[edit]
The purpose of Chapter 11, "Combining Functions" and "Power Functions", is to prepare students for 12.1 "Polynomials". Specifically, understanding that polynomials are a combination of power functions will help students to understand why we only examine the highest degree term of a polynomial when finding its end behavior.
The purpose of 11.1 is for students to 1) recognize that they have been combining functions all along and 2) develop their word problem solving skills, while applying their knowledge of function transformations and quadratic functions.
Be prepared to spend most of the lesson on group work.
Lesson Guide[edit]
Warm-Up[edit]
Have students do Problem 1.
Emphasize the importance of using parentheses. You may also need to go over how to use factoring to simplify in part (f).
Find formulas for combinations of functions[edit]
Students have seen a cost, revenue, profit word problem before in 5.3 "Linear Functions" and also have lots of experience with building simple linear formulas, so you may not need to do an example for your students. However, if you think that they need a refresher, you may want to address finding profit based on a revenue and a cost function.
Optional: Do a profit example, where you build separate functions for cost and revenue.
- Example:
Have students do Problems 2-4. If you choose not to do a profit example, then it is worth mentioning when discussing Problem 2 with your students that $P(h) = 50h$, because more generally, $P(h) = R(h) - C(h) = 250h - 200h = 50h$.
Since students tend to struggle with Problem 5, it may be helpful to do another example like Problem 5 where the combination of functions is done via multiplication and show students how considering the units of the combined functions it may help to interpret it in real world terms.
- Example:
You own a restaurant and have 5 waiters/waitresses whom you'll pay $15/hour. However, if you are understaffed any given night, you'll pay $1.25/hour extra for each missing waiter/waitress. Write a formula $f(x)$ which gives the number of waiters/waitresses working on an evening where $x$ waiters/waitresses are missing. Write a formula $g(x)$ which gives the hourly wages per waiter/waitress working on an evening where $x$ waiters/waitresses are missing.
$f(x) = 5-x$ $g(x) = 15+1.25x$
What does $f(x)g(x)$ represent? Emphasize that the meaning is in the units.
$f(x)g(x) = (5-x)(15+1.25x)$. If students are taking Chemistry, tell them that it's helpful to use dimensional analysis. We have (waiters)(wages/waiter) = wages, so $f(x)g(x)$ is how much money you need to pay all your waiters/waitresses on an evening where $x$ waiters/waitresses are missing.
Have students do Problems 5 and 6.
