Horizontal Line Test Math Example 1

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

Example 1

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

Solution

  1. 1
    The graph of f(x)=2x+5f(x) = 2x + 5 is a straight line with slope 22 (nonzero).
  2. 2
    Any horizontal line y=cy = c intersects a non-horizontal line at most once.
  3. 3
    Since every horizontal line crosses the graph at most once, ff passes the horizontal line test and is one-to-one.

Answer

Yes, f(x)=2x+5 is one-to-one.\text{Yes, } f(x) = 2x + 5 \text{ is one-to-one.}
The horizontal line test states: a function is one-to-one if and only if no horizontal line intersects its graph more than once. One-to-one functions have inverses. All linear functions with nonzero slope are one-to-one.

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 2 medium

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

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.

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