Finding and Interpreting Inverse Functions

5 min

Narrative

Earlier in the course, students have spent some time writing equations in one variable, two variables, and multiple variables to represent relationships and constraints. They have also rearranged equations to isolate a particular variable. This Warm-up activates that prior knowledge, preparing students to write equations for linear functions that involve two or more operations, and then to write their inverses.

To ensure access, the equations students are expected to write are scaffolded—starting with one operation and moving to two operations, from using numbers and letters to using only letters. The scaffolding also allows students to see structure in the equations, which would be helpful when they try to reverse the process.

Launch

Tell students that, for this activity, shipping is considered a flat fee that is charged regardless of the number of books ordered.

Student Task

Lin is comparing the cost of buying cookbooks at different online stores.

  • Store A sells them at $9 each and offers free shipping.
  • Store B sells them at $9 each and charges $5 for shipping. 
  • Store C sells them at pp dollars and charges $5 for shipping.
  • Store D sells them at pp dollars and charges ff dollars for shipping.
  1. For each store, write an equation to represent the total cost, TT, in dollars as a function of nn cookbooks bought.
  2. For each store, write an equation to find the number of books, nn, that Lin could buy if she spent a total of TT dollars.

Sample Response

    • Store A: T=9nT = 9n
    • Store B: T=9n+5T = 9n + 5
    • Store C: T=pn+5T = pn + 5
    • Store D: T=pn+fT = pn + f
    • Store A: n=T9n = \dfrac {T}{9}
    • Store B: n=T59n = \dfrac {T - 5}{9}
    • Store C: n=T5pn = \dfrac {T - 5}{p}
    • Store D: n=Tfpn = \dfrac {T-f}{p}
Activity Synthesis (Teacher Notes)

Invite students to share their equations. Record and display them for all to see. Ask students who wrote the second set of equations how they went about doing so. Highlight two main points:

  • Writing the second set of equations involves undoing each operation in the first equation (doing the operations in reverse order).
  • Each equation in the second set is the inverse of the corresponding equation in the first question. Each isolated variable is now an output, while previously it was an input. The variable that was previously isolated is now an input.
Standards
Building On
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • A-CED.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <em>For example, rearrange Ohm's law V = IR to highlight resistance R.</em>
  • HSA-CED.A.4·Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <span>For example, rearrange Ohm's law <span class="math">\(V = IR\)</span> to highlight resistance <span class="math">\(R\)</span>.</span>
Addressing
  • F-BF.4·Find inverse functions.
  • F-BF.4·Find inverse functions.
  • F-BF.4·Find inverse functions.
  • F-BF.4·Find inverse functions.
  • F-BF.4·Find inverse functions.
  • HSF-BF.B.4·Find inverse functions.

10 min

20 min