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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

Why it matters: Inverse relationships show up in logarithms, unit conversions, solving formulas, and undoing a process.

Worked example

Problem. A graph passes the horizontal line test when:

  1. The horizontal line test checks one-to-one behavior.
  2. At most one hit means each output came from at most one input.
  3. Then the inverse relation is also 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 exactly once · no horizontal line touches it

Show solution
  1. The horizontal line test checks one-to-one behavior.
  2. At most one hit means each output came from at most one input.
  3. Then the inverse relation is also a function.

Answer: every horizontal line hits it at most once

2. If one horizontal line crosses a graph three times, the function is:

Choices: not one-to-one · not a function · always linear · missing a domain

Show solution
  1. Three hits mean one output value comes from three different inputs.
  2. Different inputs sharing an output breaks one-to-one behavior.
  3. The graph can still pass the vertical line test, so it is still a function.

Answer: not one-to-one

3. A horizontal line drawn at height y = 5 crosses the parabola y = x^2 in how many points?

Show solution
  1. Solve x^2 = 5 for the crossings.
  2. Since 5 > 0, there are two solutions: x = sqrt(5) and x = -sqrt(5).
  3. So the horizontal line meets the parabola in 2 points.

Answer: 2

4. A horizontal line drawn at height y = 6 crosses the curve y = x^3 in how many points?

Show solution
  1. Solve x^3 = 6 for the crossings.
  2. A cube root gives exactly one real solution, x = cbrt(6).
  3. So the horizontal line meets the cubic in just 1 point.

Answer: 1

5. Passing the horizontal line test is exactly the same as a function being:

Choices: one-to-one (so it has an inverse function) · even · increasing everywhere then decreasing · a polynomial

Show solution
  1. The horizontal line test is the graphical test for one-to-one.
  2. One-to-one means each output traces back to a single input.
  3. That is precisely the condition for an inverse function to exist.

Answer: one-to-one (so it has an inverse function)

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