Cogito
Algebra 2 · Chapter 1 · Lesson 3
Composition and Inverse Functions
Feed one function into another, then undo it.
12 problems · about 22 minutes · F-BF.A.1, F-BF.B.4
What this lesson teaches
The student composes two functions and finds an inverse, verifying it by composition.
- f(g(x)) means do g first, then f, and the order matters.
- To find an inverse, swap x and y, then solve for y.
- Verify by composing both ways; both must return x.
Warm Up
Straightforward practice. Get the method working first.
5 problemsf(x) = x − 7. What is f⁻¹(3)?
Answer 10
Why 10.
Why does y = x² have no inverse over all real numbers?
Answer Two inputs share an output, so nothing can send it back to both.
Why 3 and −3 both give 9, so the reverse is not a function.
f(x) = x + 5. What is f⁻¹(12)?
Answer 7
Why The inverse subtracts 5.
f(x) = 4x. What is f⁻¹(20)?
Answer 5
Why The inverse divides by 4.
f(x) = 2x − 1 and g(x) = x². What is f(g(3))?
Answer 17
Why g(3) = 9, then 2(9) − 1.
Build It Up
The same ideas with more to keep track of.
3 problemsSame functions. What is g(f(3))?
Answer 25
Why f(3) = 5, then square it.
f(x) = 3x + 6. What is f⁻¹(0)?
Answer -2
Why (0 − 6) ÷ 3.
f(f⁻¹(9)) equals what?
Answer 9
Why They undo each other.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the steps for finding an inverse in order.
Answer 1. Write the function as y = something. 2. Swap x and y. 3. Solve the new equation for y. 4. Verify by composing in both directions.
Why The swap comes before the solving.
Which functions have an inverse over all real numbers?
Answer f(x) = 5x − 2; f(x) = x³
Why Two inputs sharing one output rules a function out.
The Composition: f(x) = 2x and g(x) = x + 3. What is f(g(4))?
Answer 14
Why 14.
The Other Order: With the same functions, what is g(f(4))?
Answer 11
Why 11, which differs from 14. Order matters.