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Video: Stating Whether a Given Graph Represents a Function or Not

Alex Cutbill

Does the given graph represent a function?


Video Transcript

Does the given graph represent a function?

We can see that the figure shows a line parallel to the 𝑥-axis. The question is, is this the graph of some function? We can test whether some curve drawn on the coordinate plane is the graph of a function by using the vertical line test.

Let’s draw some vertical lines on the coordinate plane, that is, lines which are parallel to the 𝑦-axis. The vertical line test says that a curve is a graph of a function if we can’t draw a vertical line which intersects it more than once. We can see that the leftmost vertical line intersects the curve that we started with only once, and the same is true for the middle vertical line; there’s only one intersection. And the same is true for the rightmost vertical line.

Looking at the figure, we can see that there’s no way we could draw a vertical line which intersects this curve more than once. And so the answer is yes, this graph represents a function. The statement of the vertical line test that we used is on the screen for you to read if you’d like to.

It’s important to understand why the vertical line test works. Imagine that you wanted to find the value of 𝑓 of four for this function. You look at the value of four on the 𝑥-axis; you go up from this point until you hit the curve; the value of the function is then the 𝑦-coordinate of this point, which you can get by reading across to the 𝑦-axis. And we see then that 𝑓 of four is five.

This process works because it was clear which point we had to go up to from the 𝑥-axis. We can’t find another point on the graph by going further up, nor could we find a point on the graph by going down from the 𝑥-axis. In other words, the vertical line 𝑥 equals four intersected this curve in only one point, this point.

However, there are curves for which this isn’t true. Which of these three values is 𝑓 of four? Well we can’t tell. This curve fails the vertical line test because the line 𝑥 equals four intersects the curve at three points. The vertical line 𝑥 equals 10 only intersects the curve at one point, but this doesn’t matter; we only have to find one vertical line which intersects the curve at more than one point for the curve to fail the vertical line test and hence for the curve not to represent a function.