# Horizontal Line Test

The **horizontal line test** is a convenient method that can determine if the inverse of a function is also a function.

It is possible that the inverse of a function is not a function because it doesn’t pass the vertical line test.

So here’s the deal!

If the horizontal line intersects the graph of a function in all places **at exactly one point**, then the given function has an *inverse that is also a function*. We say this function **passes** the horizontal line test.

Here are some examples of functions that pass the horizontal line test:

## Horizontal Line Cutting or Hitting the Graph at Exactly One Point

- Graph of the line [latex]f\left( x \right) = – x + 2[/latex].

- Graph of the square root function [latex]f\left( x \right) = \sqrt {x + 1}[/latex]

- Graph of the rational function [latex]\large{f\left( x \right) = {1 \over {x + 1}}}[/latex].

On the other hand, if the horizontal line can intersect the graph of a function in some places at **more than one point**, then the function involved can’t have an inverse that is also a function.

We say this function **fails** the horizontal line test.

Here are some examples of functions that **fail** the horizontal line test:

## Horizontal Line Cutting or Hitting the Graph at More Than One Point

- Graph of the parabola [latex]f\left( x \right) = {x^2} – 2[/latex].

- Graph of absolute value function [latex]f\left( x \right) = \left| x \right|[/latex]

- Graph of semi-circle [latex]f\left( x \right) = \sqrt {7 – {x^2}}[/latex]

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