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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 f inverse(x) in terms of x.

  1. Worked Example: First identify exactly what the question is asking: Find the inverse of f(x) = 3x + 2. Enter f inverse(x) 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: subtract 2, then divide by 3, giving 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 f inverse(x) 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 f inverse(x) 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: subtract 2, then divide by 3, giving 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) = (x - 3)/4. Enter f inverse(x) in terms of x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the inverse of f(x) = (x - 3)/4. Enter f inverse(x) 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 - 3)/4.
  4. Swap x and y: x = (y - 3)/4.
  5. Multiply by 4 and add 3: y = 4x + 3.
  6. Check the result by substituting or estimating: the response should match 4x + 3 and make sense in the original problem.

Answer: 4x + 3

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the inverse of f(x) = x + 4. Enter f inverse(x) 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 + 4.
  4. Swap x and y: x = y + 4.
  5. Solve for y: y = x - 4.
  6. Check the result by substituting or estimating: the response should match x - 4 and make sense in the original problem.

Answer: x - 4

4. Find the inverse of f(x) = x - 5. Enter f inverse(x) 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. Enter f inverse(x) 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.
  4. Swap x and y: x = y - 5.
  5. Solve for y: y = x + 5.
  6. Check the result by substituting or estimating: the response should match x + 5 and make sense in the original problem.

Answer: x + 5

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the inverse of f(x) = 3x. Enter f inverse(x) 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.
  4. Swap x and y: x = 3y.
  5. Divide by 3: y = x/3.
  6. Check the result by substituting or estimating: the response should match x/3 and make sense in the original problem.

Answer: x/3

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