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Composition and Inverse Functions

A free College Algebra lesson from the “Functions” unit, with a worked example and practice problems including step-by-step solutions.

Composition uses one function's output as another function's input. Inverse functions reverse the input-output relationship of the original function.

What you'll learn

Why it matters: Checkout totals after discount and tax, unit conversions, encryption/decryption, and formula rearranging all use composition and inverse thinking. One rule feeds the next, while an inverse reverses the original steps in order.

Worked example

Problem. If f(x) = 2x + 1 and g(x) = x - 4, find f(g(9)).

  1. Evaluate inside first: g(9) = 5.
  2. Now evaluate f(5).
  3. 2(5) + 1 = 11.

Answer: 11

Practice problems

1. If f(x) = x + 2 and g(x) = 3x, find f(g(4)).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = x + 2 and g(x) = 3x, find f(g(4)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. g(4) = 12, then f(12) = 14.
  4. Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.

Answer: 14

2. The inverse of f(x) = x - 8 is...

Choices: f^-1(x) = x + 8 · f^-1(x) = x - 8 · f^-1(x) = 8x · f^-1(x) = x/8

Show solution
  1. Core Practice: First identify exactly what the question is asking: The inverse of f(x) = x - 8 is...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Undo subtracting 8.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: f^-1(x) = x + 8

3. The inverse of f(x) = 4x + 1 is...

Choices: f^-1(x) = (x - 1)/4 · f^-1(x) = (x + 1)/4 · f^-1(x) = 4x - 1 · f^-1(x) = x/4 + 1

Show solution
  1. Challenge: First identify exactly what the question is asking: The inverse of f(x) = 4x + 1 is...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Swap x and y, then solve.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: f^-1(x) = (x - 1)/4

4. If f(x) = x - 5 and g(x) = 2x + 1, find f(g(6)).

Show solution
  1. Composition: First identify exactly what the question is asking: If f(x) = x - 5 and g(x) = 2x + 1, find f(g(6)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. g(6) = 13, then f(13) = 8.
  4. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

5. The inverse of f(x) = 2x - 7 is...

Choices: f^-1(x) = (x + 7)/2 · f^-1(x) = 2x + 7 · f^-1(x) = (x - 7)/2 · f^-1(x) = x/2 - 7

Show solution
  1. Inverses: First identify exactly what the question is asking: The inverse of f(x) = 2x - 7 is...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Undo subtracting 7, then undo multiplying by 2.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: f^-1(x) = (x + 7)/2

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