Horizontal Line Test Math Example 2

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

Example 2

medium
Does f(x)=∣x−2∣f(x) = |x - 2| pass the horizontal line test? Explain algebraically.

Solution

  1. 1
    Consider the horizontal line y=1y = 1: we need ∣x−2∣=1|x - 2| = 1.
  2. 2
    This gives x−2=1x - 2 = 1 or x−2=−1x - 2 = -1, so x=3x = 3 or x=1x = 1.
  3. 3
    Two different xx-values give the same yy-value, so the horizontal line y=1y = 1 intersects the graph twice.
  4. 4
    Therefore ff fails the horizontal line test and is NOT one-to-one.

Answer

No, f(x)=∣x−2∣ is not one-to-one.\text{No, } f(x) = |x - 2| \text{ is not one-to-one.}
The absolute value function creates a V-shape, so horizontal lines above the vertex intersect the graph twice. To make it one-to-one, you would need to restrict the domain to one side of the vertex (e.g., x≥2x \ge 2 or x≤2x \le 2).

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 3 medium

Which of the following functions are one-to-one: (a) [formula], (b) [formula], (c) [formula]?

Example 4 hard

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

← All Horizontal Line Test Examples Horizontal Line Test Concept