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. A function is one-to-one when:
- One-to-one looks at inputs versus outputs.
- No two distinct inputs may share an output.
- That is what lets the inverse be a function too.
Answer: different inputs always give different outputs
Practice problems
1. A function is one-to-one when:
Choices: different inputs always give different outputs · different outputs always give different inputs · every input gives more than one output · the inputs equal the outputs
Show solution
- One-to-one looks at inputs versus outputs.
- No two distinct inputs may share an output.
- That is what lets the inverse be a function too.
Answer: different inputs always give different outputs
2. Which table could come from a one-to-one function?
Choices: x: 3, 4, 5; y: 5, 7, 9 · x: 3, 4, 5; y: 5, 5, 9 · x: 3, 4, 5; y: 7, 7, 7 · x: 3, 4, 5; y: 9, 5, 9
Show solution
- One-to-one means no output repeats for different inputs.
- Only x: 3, 4, 5; y: 5, 7, 9 lists three different outputs.
- Any repeated output would break one-to-one.
Answer: x: 3, 4, 5; y: 5, 7, 9
3. Why is f(x) = x^2 not one-to-one over 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
- Core Practice: First identify exactly what the question is asking: Why is f(x) = x^2 not one-to-one over all real numbers?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Look for two inputs sharing an output.
- 2 and -2 both square to 4.
- Two inputs, one output, so it is not one-to-one.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: f(2) and f(-2) are both 4
4. Restricting f(x) = x^2 to x >= 0 makes it one-to-one because:
Choices: each output now comes from only one input · the graph disappears · the outputs become negative · it is no longer a function
Show solution
- The duplicate inputs were the negative ones.
- Cutting them off leaves one input per output.
- So the restricted function is one-to-one.
Answer: each output now comes from only one input
5. Which of these functions is one-to-one over all real numbers?
Choices: f(x) = 3x + 1 · f(x) = x^2 + 1 · f(x) = |x| + 1 · f(x) = 1
Show solution
- A nonzero-slope line never repeats an output.
- Parabolas and absolute values fold back and repeat outputs.
- So f(x) = 3x + 1 is the one-to-one choice.
Answer: f(x) = 3x + 1
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