Successive Discounts as a Composition of Functions

This activity demonstrates how composition of functions can be seen in the process of successive discounts. Part of the activity involves using the graphing calculator to visually see what happens as the successive discounts are applied. This activity could be used for multiple percentage discounts as well as a percentage discount connected with a purely numerical discount.

Weblink address (URL):

Standards & Objectives

Learning objectives:

Students will:

Determine numeric and symbolic representations for composition of functions.
Determine equivalent expressions for composition of functions.
Use technology to develop graphical representations for composition of functions.

Essential and guiding questions:

Compare and contrast the algebraic representations for r(x) and s(x). How does the order of the discounts affect the algebraic representations?
How did you determine an appropriate domain for the problem situation? (This is an interesting question to explore, because if the jeans are cheap enough, it is possible to have a negative price after both discounts. One thing to discuss is that the store would not pay the customer, so if the discounts would result in a negative price, the item would just be free.)
What is the significance of the point of intersection of the graphs of y = f(x) and y = g(x) as related to the problem situation?
How did you determine functions equivalent to y = r(x) and y = s(x)?
How do the graphs of y = r(x) and y = s(x) illustrate that the sale prices represented by the function y = s(x) will always be greater than the sale prices represented by the function y = r(x)?

Lesson Variations

Blooms taxonomy level:

Understanding

Extension suggestions:

Incorporate a third discount, such as save an additional 10% by using the store credit card. Have students determine compositions of three functions.
Provide students with composite functions such as h(x) = 0.6(x – 2). Have students determine functions, f(x) and g(x), such that h(x) = f(g(x)).

Helpful Hints

Materials:

Successive Discounts Activity Sheet
Graphing Calculators

References

Contributors:

Eric Istre

Originator:

Ayers Institute for Learning & Innovation