Inverse Functions Basics
A free Algebra II lesson from the “Functions and Notation” unit, with a worked example and practice problems including step-by-step solutions.
An inverse function reverses the action of another function. If a function sends an input to an output, the inverse sends that output back to the original input. This idea matters because inverses help solve equations and connect operations such as squaring and square rooting, or exponentials and logarithms. When practicing, think about undoing operations in the opposite order. For symbolic inverses, replace f(x) with y, switch x and y, then solve for y. A common mistake is reversing only one operation instead of undoing the whole rule step by step.
What you'll learn
- Interpret inverse notation
- Find simple inverse rules
- Check inverse outputs
Worked example
Problem. Find the inverse of f(x) = 2x + 5.
- Write y = 2x + 5.
- Swap x and y: x = 2y + 5.
- Solve: y = (x - 5)/2.
Answer: f^-1(x) = (x - 5)/2
Practice problems
1. If f(x) = x + 9, find f^-1(14).
Show solution
- Warm-up: First identify exactly what the question is asking: If f(x) = x + 9, find f^-1(14).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The inverse undoes +9.
- 14 - 9 = 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
2. If f(x) = 3x, find f^-1(21).
Show solution
- Warm-up: First identify exactly what the question is asking: If f(x) = 3x, find f^-1(21).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Undo multiplying by 3.
- Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.
Answer: 7
3. The inverse of f(x) = x - 4 is...
Choices: f^-1(x) = x + 4 · f^-1(x) = x - 4 · f^-1(x) = 4x · f^-1(x) = x/4
Show solution
- Core Practice: First identify exactly what the question is asking: The inverse of f(x) = x - 4 is...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Undo subtracting 4 by adding 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 + 4
4. The inverse of f(x) = 5x is...
Choices: f^-1(x) = x/5 · f^-1(x) = x - 5 · f^-1(x) = x + 5 · f^-1(x) = 5x
Show solution
- Core Practice: First identify exactly what the question is asking: The inverse of f(x) = 5x is...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Undo multiplying by 5.
- 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/5
5. The inverse of f(x) = 2x - 6 is...
Choices: f^-1(x) = (x + 6)/2 · f^-1(x) = (x - 6)/2 · f^-1(x) = 2x + 6 · f^-1(x) = x/2 - 6
Show solution
- Challenge: First identify exactly what the question is asking: The inverse of f(x) = 2x - 6 is...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Swap x and y, then solve.
- 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 + 6)/2
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