Finding an Inverse: Algebraic Unpacking
Many textbooks and online resources teach students to find an inverse using the
following steps:
1.
2.
3.
4.
Replace f (x) with y.
Reverse the roles of x and y.
Solve for y in terms of x.
Replace y with f 1 (x) .
While this method helps to develop procedural fluency, it does not provide students
with a deep understanding of what an inverse is or why they are performing this
procedure.
Algebraic Unpacking is a conceptual method for determining the inverse of a
function. Before applying the strategy to specific functions, first have students
review what they currently know about inverse operations (from Algebra I and
Middle School Mathematics). Students should be able to identify concepts, such as
“undoing”, “working backwards” or “retracing steps to return to an original value”.
Sample Actions & Inverse Actions:
Open the door ------------------------------------------ Close the door
Turn on a light ----------------------------------------- Turn off the light
Add 5 to a number ------------------------------------- Subtract 5 from a number
Multiply a number by 2, then subtract 3 ---------- Add 3 to a number, then divide by 2
As a visual, you can show the steps for wrapping a present and then have students
explain what the steps would be to unwrap a present.
Then explain that inverses/inverse functions are formulas that “undoes” the original
function. To determine the inverse of a function, you need to derive the function
that “undoes” the original function.
Linear Example: Find the inverse for f (x)  3x  4 .
For any number x, you need to first multiply the number by 3 and then subtract 4.
The inverse actions would be to first add 4 to a number and then divide by 3.
x4
So, the inverse is f 1 (x) 
.
3
x3
2.
7
For any number x, you need to first raise the number to the third power, then divide
by 7, then add 2. The inverse actions would be to first subtract 2 from a number,
then multiply by 7, then find the cube root of the result.
So, the inverse is g1 (x)  3 7(x  2)
Cubic Example: Find the inverse for g(x) 