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Cogito

Algebra 2 · Chapter 1 · Lesson 3

Practice Worksheet

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

NameDate

Figure — use these to answer the problems

A line and its inverse are mirror images across y = x. Change the slope and watch the reflection.
-10-10-5-5551010
y = 2x + 1
Slope2
y-intercept1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. f(x) = x − 7. What is f⁻¹(3)?

    Answer

  2. Why does y = x² have no inverse over all real numbers?

    • Two inputs share an output, so nothing can send it back to both.
    • Its graph is a curve.

  3. f(x) = x + 5. What is f⁻¹(12)?

    Answer

  4. f(x) = 4x. What is f⁻¹(20)?

    Answer

  5. f(x) = 2x − 1 and g(x) = x². What is f(g(3))?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Same functions. What is g(f(3))?

    Answer

  2. f(x) = 3x + 6. What is f⁻¹(0)?

    Answer

  3. f(f⁻¹(9)) equals what?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the steps for finding an inverse in order.

    Write 1 to 4 in the boxes to put these in order.

    • Write the function as y = something.
    • Swap x and y.
    • Solve the new equation for y.
    • Verify by composing in both directions.

  2. Which functions have an inverse over all real numbers?

    • f(x) = 5x − 2
    • f(x) = x³
    • f(x) = x²
    • f(x) = |x|
    • Tick every box that applies.

  3. The Composition

    f(x) = 2x and g(x) = x + 3. What is f(g(4))?

    Answer

  4. The Other Order

    With the same functions, what is g(f(4))?

    Answer