Horizontal Line Test Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Which of the following functions are one-to-one: (a) f(x)=x3f(x) = x^3, (b) g(x)=x3−3xg(x) = x^3 - 3x, (c) h(x)=exh(x) = e^x?

Solution

  1. 1
    (a) f(x)=x3f(x) = x^3 is strictly increasing on all reals, so it is one-to-one. (c) h(x)=exh(x) = e^x is strictly increasing, so it is one-to-one.
  2. 2
    (b) g(x)=x3−3xg(x) = x^3 - 3x: g′(x)=3x2−3=0g'(x) = 3x^2 - 3 = 0 at x=±1x = \pm 1. g(−1)=2g(-1) = 2, g(1)=−2g(1) = -2. The horizontal line y=0y = 0 gives x3−3x=0x^3 - 3x = 0, x(x2−3)=0x(x^2-3) = 0, so x=0,±3x = 0, \pm\sqrt{3} — three intersections. Not one-to-one.

Answer

(a) and (c) are one-to-one; (b) is not\text{(a) and (c) are one-to-one; (b) is not}
A function is one-to-one if and only if it is strictly monotonic (always increasing or always decreasing). Functions with local maxima and minima fail the horizontal line test because there exist horizontal lines that cross the graph multiple times.

About Horizontal Line Test

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.

Learn more about Horizontal Line Test →

More Horizontal Line Test Examples

Example 1 easy

Use the horizontal line test to determine whether [formula] is one-to-one.

Example 2 medium

Does [formula] pass the horizontal line test? Explain algebraically.

Example 4 hard

Prove algebraically that [formula] (for [formula]) is one-to-one without graphing.

← All Horizontal Line Test Examples Horizontal Line Test Concept