One-to-One Functions
A free Precalculus lesson from the “Inverse Functions” unit, with a worked example and practice problems including step-by-step solutions.
A one-to-one function never sends two different inputs to the same output. 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
- Decide whether a function is one-to-one from pairs, tables, or rules
- Use one-to-one functions in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Which table could be one-to-one?
- One-to-one means different inputs have different outputs.
- Only the first table has no repeated output for different inputs.
- A repeated output would make the inverse fail the function rule.
Answer: x: 1, 2, 3; y: 4, 5, 6
Practice problems
1. Which table could be one-to-one?
Choices: x: 1, 2, 3; y: 4, 5, 6 · x: 1, 2, 3; y: 4, 4, 6 · x: 1, 2, 3; y: 7, 7, 7 · x: 1, 1, 2; y: 3, 4, 5
Show solution
- One-to-one means different inputs have different outputs.
- Only the first table has no repeated output for different inputs.
- A repeated output would make the inverse fail the function rule.
Answer: x: 1, 2, 3; y: 4, 5, 6
2. Which table could be one-to-one? (variation 2)
Choices: x: 1, 2, 3; y: 4, 5, 6 · x: 1, 2, 3; y: 4, 4, 6 · x: 1, 2, 3; y: 7, 7, 7 · x: 1, 1, 2; y: 3, 4, 5
Show solution
- One-to-one means different inputs have different outputs.
- Only the first table has no repeated output for different inputs.
- A repeated output would make the inverse fail the function rule.
Answer: x: 1, 2, 3; y: 4, 5, 6
3. Why is f(x) = x^2 not one-to-one on all real numbers?
Choices: f(2) and f(-2) are both 4 · it has no outputs · it is a polynomial · it has a y-intercept
Show solution
- Two different inputs can give the same output.
- 2 and -2 both square to 4.
- So the inverse relation would repeat an input.
Answer: f(2) and f(-2) are both 4
4. Why is f(x) = x^2 not one-to-one on all real numbers? (variation 2)
Choices: f(2) and f(-2) are both 4 · it has no outputs · it is a polynomial · it has a y-intercept
Show solution
- Two different inputs can give the same output.
- 2 and -2 both square to 4.
- So the inverse relation would repeat an input.
Answer: f(2) and f(-2) are both 4
5. Restricting the domain of a function can help it become:
Choices: one-to-one · a fraction · a vertical shift · a constant output
Show solution
- Some functions fail one-to-one because the domain is too wide.
- Restricting the domain can remove duplicate outputs.
- This is common with square-root inverses of quadratics.
Answer: one-to-one
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