What is the Vertical Line Test?

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The concept of a function is a key one in mathematics. It's an operation that relates elements from an input set, called the domain, to elements in an output set, which is called the range. Mathematicians commonly explain functions by comparing them to machines, such as a penny stamping machine. When you input a penny, the machine performs an operation, and a stamped souvenir emerges. Like a penny stamping machine, a function relates each input element to one and only one output element. If you express the relationship as a graph, a vertical line intersecting the horizontal axis at any point can pass through only one point of the graph. If it passes through more than one point, the relationship is not a function.

What Does a Function Look Like?

You can express a function simply as a set of points, but you'll usually see it in the form f(​x​) equals some relationship of ​x​. For example:

f(x) = x^2

Sometimes, another letter is used for f(​x​), most commonly ​y​. For example​:

y = x^2

The choice of letters is not important.

T = m^2 + m + 1

is also a function.

To qualify as a function, a relationship must relate each element in the domain to one and only one element in the range. For example,

f(x) = \big((2, 3), (4 ,6)\big)

is a function, but

g(x) = \big((3, 4), (3 , 9)\big)

is not.

Using the Vertical Line Test

To use the vertical line test, you have to be able to graph the relationship. This is easy if you have a set of points. You simply plot them on a set of coordinate axes. If you have an equation, you get a point set by inputting various values and recording the outputs. Once you have the set, you plot the points and draw a graph.

After drawing the graph, imagine a vertical line at the far left of the horizontal axis and move it to the right. If the line intersects more than one point in the curve at any place along its journey on the axis, the graph does not represent a function.

What Is the Horizontal Line Test?

After you have graphed a relationship and used the vertical line test to determine that it's a function, you can conduct the horizontal line test to determine whether or not it's a one-to-one function. This means that every element of the range corresponds to only one element in the domain. A straight line is an example of a one-to-one function, but a parabola is not, because every input value produces two solutions in the range.

To use the horizontal line test, imagine a horizontal line at the top of the vertical axis. Move it down the axis, and if it touches more than one point at any place along its journey, the function is not one-to-one.