Vertical Line Test

The vertical line test is used to determine if a given graph represents a function. The vertical line test states that any given vertical line can only intersect the graph of a function, at most, at one point for it to represent a function. If the graph of the equation is cut by the vertical line at more than one point for any possible vertical line, the graph can not represent a function.

The graph below does not represent a function because the vertical line, $x=1$ , intersects the graph shown in red at 3 points.

Next, let's consider the graph of the function $f\left(x\right)=\frac{1}{x}$ . This passes the vertical line test because there is no vertical line we can draw that intercepts the graph more than once.

Consider the piecewise graph below.

You can see that no vertical line passes through two points on the graph.

Note that $\left(-2,0\right)$ and $\left(1,-2\right)$ are open points and are not on the graph. So the vertical line $x=-2$ and $x=1$ passes through only one point on each of the line segments.

Therefore, the graph shown is a function.

The piecewise function can be defined as:

$y=\frac{1}{2}x+1$

-6 less than or equal to $x<-2$

$y=-x–1$

-2 less than or equal to $x<1$

$y=5$

1 less than or equal to x less than or equal to 6

The domain of the function is -6 less than or equal to x less than or equal to 6 and its range is -2 less than or equal to y less than or equal to 5.

How to apply a vertical line test

The following steps are necessary to apply the vertical line test to see if the given expression is a function or not.

Geometrically

Draw the graph of $y=f\left(x\right)$ , with respect to the coordinate axis. Now draw the line x = a, and observe the number of places it cuts the curve $y=f\left(x\right)$ . If the vertical line cuts the curve at more than one point, it does not represent a function. If it cuts the curve one time only, it represents a function.

Algebraically

The equation of a vertical line is $x=a$ . Substituting it in the equation of a curve $y=f\left(x\right)$ , we get $y=f\left(a\right)$ . If we get more than one value for y, then it proves the equation $y=f\left(x\right)$ does not represent a function. On the other hand, if we get only a single value, or no value for y, by substituting $x=a$ in $y=f\left(x\right)$ for all x, then $y=f\left(x\right)$ represents a function.

Topics related to the Vertical Line Test

One to One Functions

y-Intercepts

Range (of a Function)

Flashcards covering the Vertical Line Test

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Vertical Line Test

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

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