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 f\left( x \right) = - x + 2.
the graph of the line f(x)=-x+2 is cut by a horizontal at exactly one point
  • Graph of the square root function f\left( x \right) = \sqrt {x + 1}

the graph of an square root function f(x)=sqrt(x=1) is cut by a horizontal line at exactly one point.
  • Graph of the rational function \large{f\left( x \right) = {1 \over {x + 1}}}.

the graph of a rational function f(x)=1/(x+1) is cut by the horizontal line at exactly one point

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.
the graph of a quadratic function is cut by horizontal line in exactly two points
  • Graph of absolute value function f\left( x \right) = \left| x \right|

the graph of the absolute value function is cut by a horizontal line at two points
  • Graph of semi-circle f\left( x \right) = \sqrt {7 - {x^2}}

the graph of a semi-circle is cut by a horizontal line at exactly two points

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Vertical Line Test