## Is This Relation a Function?

This post about whether a relation is a function is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

## Question

One or more of the graphs below represent 𝑦 as a function of 𝑥. Which one or ones?

## Solution

Function/not function?

A function is a relation – that is, a correspondence between *x* and *y* – for which every *x*-value relates to only one *y*-value. No vertical line will pass through more than one point on a function’s graph, but a vertical line may pass through more than one point on the graph of a relation that is not a function. Thus you can use the vertical line test on a relation to see whether it is a function.

Look at relation A. Imagine sliding a vertical line across it. At no point will the line intersect the relation more than once. This relation **is a function.**

Now look at relation B. The *y*-axis, which is a vertical line, intersects the relation in three places. Thus relation B fails the vertical line test and relation **B is not a function.**

Relation C: Again, there are places where you could draw a vertical line and cross the relation more than once. For example, the y-axis crosses the relation twice. This relation is **not a function.**

Relation D: There is only one *x*-value where it looks like a vertical line could cross this relation twice: the *x*-value at the right-hand end of the lower piece, which is the *x*-value at left-hand end of the upper piece. Are there two *y*-values at this *x*-value? The lower piece, in which the circle is filled in, does take this *x*-value. But the upper piece, in which the circle is not filled in, does not take the value. Thus there is no *x*-value that both pieces take, and there is no *x*-value for which there is more than one value of **This relation is a function**.

𝑨𝒏𝒔𝒘𝒆𝒓: **A and D**

This question is similar to #6 in the sample questions for the Accuplacer Advanced Algebra and Functions test.