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Video: Determining Whether Relations Are Functions

Bethani Gasparine

Which of the relations shown by the set of the ordered pairs below does not represent a function?

02:33

Video Transcript

Which of the relations shown by the set of ordered pairs below does not represent a function?

In order for a relation to be a function, then every input value, every 𝑥-value, has to have exactly one output value, one 𝑦-value that goes with it. A function that we actually know would be a quadratic function, for example. We can have a linear function, a quadratic, a cubic function.

Let’s think of a quadratic, 𝑦 equals 𝑥 squared. Say we plug in two for 𝑥, two squared is equal to four. Now if I would plug in two again, I’m still gonna get four. So the input value 𝑥, there’s exactly one output value. It’s only four. Now it is possible to plug in a different input value, such as if we plugged in negative two. Negative two squared is also positive four. So it’s okay to have the 𝑦s repeating, but we don’t want the 𝑥s to repeat with different 𝑦-values, such as having two, four and two, eight. That doesn’t make sense. When you plug in two, you should get one answer.

So looking at option A), we plug in negative eight, we get four. We plug in negative nine, we get four. And we plug in ten and eleven, and we still get four. That is a function. It’s actually the function — it’s the horizontal line 𝑦 equals four. And graphically, you can see that this is a function by the vertical line test. That means if you would draw a vertical line anywhere on that graph and it only touches once, it’s a function. So here, we can see a couple more vertical lines, and again they only touch once.

Option B), we plug in three, we get four. We plug in eleven, we get eight. Now we plug in three again, and this time we get a different answer; we get twelve. That doesn’t make sense. So this means B) does not represent a function because we plug in three and we get four, and then we plug in three again and we get a different answer of twelve. So if we think of those two points on a graph, that would not pass the vertical line test. At that point, we would actually cross the vertical line twice, which would make it not a function.

So again, B) is the relation that is not a function. And this is because the input value 𝑥, the input value of three, did not have exactly one output value. It didn’t have one 𝑦 value. It had two; it had four and twelve.