### Video Transcript

Is this the graph of a linear or a nonlinear function?

In order to identify whether this is the graph of a linear or a nonlinear function, let’s begin by reminding ourselves what we mean when we describe an equation as a function. A function is a rule that relates an element in one set to exactly one element of a second set. We talk about functions as being one to one, in other words, one element in the first set maps to exactly one in the second, or many to one. That is, several elements in the first set could map to exactly one in the second. A one-to-many relationship is not a function. For this reason, a vertical line cannot represent a function. This would be an example of a one-to-many relationship. We substitute one value in, and we get a whole bunch out.

So with that in mind, let’s remind ourselves what we mean by linear and nonlinear functions. A linear function is a function whose graph is a straight line. Therefore, a nonlinear function has a graph which is not a straight line. An example of this is something like 𝑦 equals 𝑥 cubed or 𝑦 equals 𝑒 to the power of 𝑥. And if we inspect our graph, we can see it is a nonvertical straight line. It must then be the graph of a linear function. In fact, we’re able to deduce the equation of this linear function. We might know this by heart, but let’s identify a few key points that lie on this line.

We have the point with coordinates four, three; another point with coordinates negative two, three; and another with coordinates negative four, three. In fact, every single point that lies on this horizontal line has a 𝑦-coordinate of three. And so we can say that its equation is 𝑦 equals three. Or if we wish to choose function notation, we can write 𝑓 of 𝑥 equals three. This is the graph of a linear function.