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 โ€” 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)=exf(x) = e^x pass the horizontal line test?

Example 3

medium
Does f(x)=sinโกxf(x) = \sin x pass the horizontal line test on all reals?

Example 4

medium
Does f(x)=xf(x) = \sqrt{x} (for xโ‰ฅ0x \ge 0) pass the horizontal line test?

Example 5

medium
Does f(x)=(xโˆ’1)2+3f(x) = (x-1)^2 + 3 pass the test on xโ‰ฅ1x \ge 1?

Example 6

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

Example 7

medium
Which of the following functions are one-to-one: (a) f(x)=x3f(x) = x^3, (b) g(x)=x3โˆ’3xg(x) = x^3 - 3x, (c) h(x)=exh(x) = e^x?

Example 8

easy
Does f(x)=tanโกxf(x) = \tan x on (โˆ’ฯ€/2,ฯ€/2)(-\pi/2, \pi/2) pass the horizontal line test?

Example 9

hard
For f(x)=x2f(x) = x^2 restricted to [โˆ’2,1][-2, 1], does ff pass the horizontal line test?

Example 10

medium
Does f(x)=x2f(x) = x^2 restricted to xโ‰ฅ0x \ge 0 pass the horizontal line test?

Example 11

hard
For f(x)=exโˆ’eโˆ’xf(x) = e^x - e^{-x}, show ff passes the horizontal line test on all reals.

Example 12

challenge
Find the largest interval [0,k][0, k] on which f(x)=sinโกxf(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)=x3โˆ’3x+1f(x) = x^3 - 3x + 1 pass the horizontal line test on all reals?

Example 15

medium
For f(x)=x4f(x) = x^4, find the largest interval starting at 0 on which ff passes the horizontal line test.

Example 16

challenge
For f(x)=sinโกxf(x) = \sin x, find the largest interval [a,a+ฯ€][a, a+\pi] that contains 00 and on which ff passes the horizontal line test.

Example 17

easy
Use the horizontal line test to determine whether f(x)=2x+5f(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=2y = 2 cross f(x)=x2+1f(x) = x^2 + 1? Use this to comment on the horizontal line test.

Example 20

medium
Does f(x)=1/xf(x) = 1/x (for xโ‰ 0x \ne 0) pass the horizontal line test?
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