Practice Horizontal Line Test in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The horizontal line test is a visual method to determine whether a function is one-to-one (injective). If every horizontal line intersects the function's graph at most once, the function passes the test and has an inverse function on its full domain.

A horizontal line that crosses the graph at two points means those two inputs produce the same output — the function is many-to-one and has no inverse without domain restriction.

Showing a random 20 of 50 problems.

Example 1

challenge
State and justify: a continuous function on a closed interval that passes the horizontal line test must be strictly monotonic.

Example 2

easy
Does f(x)=ex pass the horizontal line test?

Example 3

medium
Does f(x)=sin⁡x pass the horizontal line test on all reals?

Example 4

medium
Does f(x)=x (for x≥0) pass the horizontal line test?

Example 5

medium
Does f(x)=(x−1)2+3 pass the test on x≥1?

Example 6

easy
Does f(x)=2x+1 pass the horizontal line test?

Example 7

medium
Which of the following functions are one-to-one: (a) f(x)=x3, (b) g(x)=x3−3x, (c) h(x)=ex?

Example 8

easy
Does f(x)=tan⁡x on (−π/2,π/2) pass the horizontal line test?

Example 9

hard
For f(x)=x2 restricted to [−2,1], does f pass the horizontal line test?

Example 10

medium
Does f(x)=x2 restricted to x≥0 pass the horizontal line test?

Example 11

hard
For f(x)=ex−e−x, show f passes the horizontal line test on all reals.

Example 12

challenge
Find the largest interval [0,k] on which f(x)=sin⁡x passes the horizontal line test.

Example 13

easy
If a graph is shaped like a sideways parabola opening right, does it pass the horizontal line test?

Example 14

medium
Does f(x)=x3−3x+1 pass the horizontal line test on all reals?

Example 15

medium
For f(x)=x4, find the largest interval starting at 0 on which f passes the horizontal line test.

Example 16

challenge
For f(x)=sin⁡x, find the largest interval [a,a+π] that contains 0 and on which f passes the horizontal line test.

Example 17

easy
Use the horizontal line test to determine whether f(x)=2x+5 is one-to-one.

Example 18

medium
A function passes the horizontal line test. What can you conclude about its inverse?

Example 19

medium
How many times does y=2 cross f(x)=x2+1? Use this to comment on the horizontal line test.

Example 20

medium
Does f(x)=1/x (for x≠0) pass the horizontal line test?
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