Horizontal Line Test Math Example 4

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

Example 4

hard
Prove algebraically that f(x)=xx+1f(x) = \frac{x}{x+1} (for x>−1x > -1) is one-to-one without graphing.

Solution

  1. 1
    Assume f(a)=f(b)f(a) = f(b): aa+1=bb+1\frac{a}{a+1} = \frac{b}{b+1}. Cross-multiply: a(b+1)=b(a+1)a(b+1) = b(a+1), so ab+a=ab+bab + a = ab + b.
  2. 2
    Subtract abab: a=ba = b. Since f(a)=f(b)f(a) = f(b) implies a=ba = b, the function is one-to-one.

Answer

Proven: f(a)=f(b)  ⟹  a=b\text{Proven: } f(a) = f(b) \implies a = b
The algebraic test for one-to-one: assume f(a)=f(b)f(a) = f(b) and show this forces a=ba = b. This is equivalent to the horizontal line test but does not require a graph. It works well for rational functions and other functions that are hard to visualize.

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 3 medium

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

← All Horizontal Line Test Examples Horizontal Line Test Concept