Cogito
AP Precalculus · Chapter 4 · Lesson 2
Inverse Functions
Undoing, and when it is allowed.
11 problems · about 22 minutes · AP Precalculus 1.15
What this lesson teaches
The student finds an inverse function and explains why one-to-one is required.
- An inverse sends each output back to the input it came from.
- Find it by swapping x and y, then solving for y.
- Only a one-to-one function has an inverse, and the graph is a reflection in y = x.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = 4x + 2. Inverse of 22?
Answer 5
Why 5.
f(x) = x − 4. Inverse of 10?
Answer 14
Why Add 4 back.
f(x) = 2x. Inverse of 18?
Answer 9
Why Halve it.
f(x) = 3x − 1. Inverse of 14?
Answer 5
Why Add 1, then divide by 3.
Build It Up
The same ideas with more to keep track of.
3 problemsSelect every function that has an inverse over all real numbers.
Answer f(x) = 2x + 1; f(x) = x³
Why Ask whether two different inputs can share an output.
Across which line is an inverse a reflection? Answer 1 for y = x, 2 for the x-axis.
Answer 1
Why Swapping coordinates.
The point (3, 11) is on f. What is the x of the matching point on the inverse?
Answer 11
Why The coordinates swap.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder the steps for finding an inverse.
Answer 1. Write y = f(x) 2. Swap x and y 3. Solve for y 4. Check with one point
Why Swap before solving.
Why has x² no inverse over all reals? Answer 1 if two inputs share an output, 0 otherwise.
Answer 1
Why 2 and −2 both give 4.
Undoing: f(x) = x + 8. What does the inverse give when the input is 20?
Answer 12
Why 12.
The Round Trip: f(x) = 5x. What is f applied to the inverse of 45?
Answer 45
Why 45 — you end where you started.