CMClearMathAcademy

Finding Inverses Algebraically

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

To find an inverse, write y = f(x), swap x and y, then solve for y. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Inverse relationships show up in logarithms, unit conversions, solving formulas, and undoing a process.

Worked example

Problem. Find the inverse of f(x) = 3x + 2. Enter in terms of x.

  1. Worked Example: First identify exactly what the question is asking: Find the inverse of f(x) = 3x + 2. Enter in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = 3x + 2.
  4. Swap x and y: x = 3y + 2.
  5. Solve for y: y = (x - 2)/3.
  6. Check the result by substituting or estimating: the response should match (x - 2)/3 and make sense in the original problem.

Answer: (x - 2)/3

Practice problems

1. Find the inverse of f(x) = 3x + 2. Enter in terms of x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the inverse of f(x) = 3x + 2. Enter in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = 3x + 2.
  4. Swap x and y: x = 3y + 2.
  5. Solve for y: y = (x - 2)/3.
  6. Check the result by substituting or estimating: the response should match (x - 2)/3 and make sense in the original problem.

Answer: (x - 2)/3

2. Find the inverse of f(x) = 4x + 3. Enter in terms of x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the inverse of f(x) = 4x + 3. Enter in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = 4x + 3.
  4. Swap x and y: x = 4y + 3.
  5. Solve for y: y = (x - 3)/4.
  6. Check the result by substituting or estimating: the response should match (x - 3)/4 and make sense in the original problem.

Answer: (x - 3)/4

3. Find the inverse of f(x) = 5x + 4. Enter in terms of x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the inverse of f(x) = 5x + 4. Enter in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = 5x + 4.
  4. Swap x and y: x = 5y + 4.
  5. Solve for y: y = (x - 4)/5.
  6. Check the result by substituting or estimating: the response should match (x - 4)/5 and make sense in the original problem.

Answer: (x - 4)/5

4. Find the inverse of f(x) = (x - 5)/2. Enter in terms of x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the inverse of f(x) = (x - 5)/2. Enter in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = (x - 5)/2.
  4. Multiply by 2: 2y = x - 5.
  5. Swap and solve to get f inverse(x) = 2x + 5.
  6. Check the result by substituting or estimating: the response should match 2x + 5 and make sense in the original problem.

Answer: 2x + 5

5. Find the inverse of f(x) = (x - 1)/3. Enter in terms of x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the inverse of f(x) = (x - 1)/3. Enter in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = (x - 1)/3.
  4. Multiply by 3: 3y = x - 1.
  5. Swap and solve to get f inverse(x) = 3x + 1.
  6. Check the result by substituting or estimating: the response should match 3x + 1 and make sense in the original problem.

Answer: 3x + 1

Practice this interactively with instant feedback and an AI tutor.

Practice Finding Inverses Algebraically Take the free placement check

More Precalculus lessons