# Horizontal Line Test

The horizontal line test is a convenient method that can determine whether a given function has an inverse, but more importantly to find out if the inverse is also a function.

Remember that it is very possible that a function may have an inverse but at the same time, the inverse 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 should have 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 $f\left( x \right) = - x + 2$.

• Graph of the square root function $f\left( x \right) = \sqrt {x + 1}$

• Graph of the rational function $f\left( x \right) = {1 \over {x + 1}}$.

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 $f\left( x \right) = {x^2} - 2$.

• Graph of absolute value function $f\left( x \right) = \left| x \right|$

• Graph of semi-circle $f\left( x \right) = \sqrt {7 - {x^2}}$

You might also be interested in:

Vertical Line Test