Interpreting Relations and the Vertical Line Test

Checking to see if a graph is a function

The vertical line test checks to see if a curve on a graph can be described as a function. Functions are defined as having at most one $y$ value for any one $x$ value, so if there's ever a place on a graph where there are two values of $y$ for one value of $x$, that curve can't be described as a function.

We can check graphs for this by taking a vertical line and passing it horizontally over the graph, looking at where the vertical line intersects with the graph. If there's ever a place where the line intersects more than once, the curve on the graph fails the test and can't be described by a function.

For example, look at the graph below:

In the graph above, curve B could represent a function, while curve A could not. If you look at the image below, you'll see that the red vertical line passes through curve A three times at the $x$-value of $4.$ At other $x$-values, like $x=2, 3,$ or $5,$ this is also the case. This means the curve fails the vertical line test, and graphs that fail the vertical line test cannot match functions because they have more than one $y$ value for some value(s) of $x.$

(Video) Relations and Functions

by yaymath

In this video, yaymath shows you how to find domain and range of a function and goes over some examples.

A function is a relationship or expression involving one or more variables. It'll produce one (and only one) value for each possible value of $x$.

The domain of a function is all the possible $x$ values that will work in that function.

The range of a function is all the possible $y$ values that function will spit out.

You can use the vertical line test to check if a graph is a function. If the vertical line touches more than one point then the graph is not a function.

The Vertical Line Test is a way to tell if a graph represents a function