Verifying a Function Using the Vertical Line Test

We can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

The most common graphs name the input value [latex]x[/latex] and the output value [latex]y[/latex], and we say [latex]y[/latex] is a function of [latex]x[/latex], or [latex]y=f(x)[/latex] when the function is named [latex]f[/latex].

The graph of the function is the set of all points [latex](x,y)[/latex] in the plane that satisfies the equation [latex]y=f(x)[/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the [latex]x[/latex]-coordinate of each point is an input value and the [latex]y[/latex]-coordinate of each point is the corresponding output value.

For example, the black dots on the graph above tell us that [latex]f(0)=2[/latex] and [latex]f(6)=1[/latex]. However, the set of all points [latex](x,y)[/latex] satisfying [latex]y=f(x)[/latex] is a curve. The curve shown includes [latex](0,2)[/latex] and [latex](6,1)[/latex] because the curve passes through those points.

The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.

Which of the graphs represent(s) a function [latex]y=f\left(x\right)?[/latex]

Show Solution

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most [latex]x[/latex]-values, a vertical line would intersect the graph at more than one point.

Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown below, the stock price was [latex]$1000[/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of [latex]$1000[/latex].

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in the table below.

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

The Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If we can draw any horizontal line that intersects a graph more than once, then the graph does not represent a one-to-one function because that [latex]y[/latex] value has more than one input.

Which of the graphs represent(s) a one-to-one function?

Show Solution

The function in (a) is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once. The function in (c) is not one-to-one. The horizontal line intersects the graph of the function at two points.