Horizontal Line Test
A free Precalculus lesson from the “Inverse Functions” unit, with a worked example and practice problems including step-by-step solutions.
If a horizontal line hits a graph more than once, the inverse relation would fail the function test. 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
- Use the horizontal line test to judge inverse-function readiness
- Use horizontal line test in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. A graph passes the horizontal line test when:
- The horizontal line test checks one-to-one behavior.
- At most one hit means each output came from at most one input.
- Then the inverse can be a function.
Answer: every horizontal line hits it at most once
Practice problems
1. A graph passes the horizontal line test when:
Choices: every horizontal line hits it at most once · every vertical line hits it twice · it crosses the y-axis · it has no x-intercepts
Show solution
- The horizontal line test checks one-to-one behavior.
- At most one hit means each output came from at most one input.
- Then the inverse can be a function.
Answer: every horizontal line hits it at most once
2. A graph passes the horizontal line test when: (variation 2)
Choices: every horizontal line hits it at most once · every vertical line hits it twice · it crosses the y-axis · it has no x-intercepts
Show solution
- The horizontal line test checks one-to-one behavior.
- At most one hit means each output came from at most one input.
- Then the inverse can be a function.
Answer: every horizontal line hits it at most once
3. A graph passes the horizontal line test when: (variation 3)
Choices: every horizontal line hits it at most once · every vertical line hits it twice · it crosses the y-axis · it has no x-intercepts
Show solution
- The horizontal line test checks one-to-one behavior.
- At most one hit means each output came from at most one input.
- Then the inverse can be a function.
Answer: every horizontal line hits it at most once
4. If one horizontal line hits a graph three times, the function is:
Choices: not one-to-one · not a function · always linear · missing a domain
Show solution
- Multiple horizontal hits mean one output has several inputs.
- That breaks one-to-one behavior.
- The original graph may still pass the vertical line test.
Answer: not one-to-one
5. If one horizontal line hits a graph three times, the function is: (variation 2)
Choices: not one-to-one · not a function · always linear · missing a domain
Show solution
- Multiple horizontal hits mean one output has several inputs.
- That breaks one-to-one behavior.
- The original graph may still pass the vertical line test.
Answer: not one-to-one
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