### Video Transcript

Which of the following is a one-to-one function? Is it a) π of π₯ equals π₯ to the fourth power plus π₯ squared, b) π of π₯ equals π₯ squared, c) π of π₯ equals cos of π₯, or d) π of π₯ equals π₯ cubed?

We begin by recalling what it actually means for a function to be one-to-one. A function is one-to-one if each element of the range of the function corresponds to exactly one element of the domain and vice versa. One way of testing for a one-to-one function is to consider the shape of the graph. If it passes the vertical line test, that is a vertical line anywhere on the graph intersects the graph exactly once, and the horizontal line test, that is a horizontal line intersects the curve exactly once. Then, we can say a function is one-to-one. And so, the easiest way to check whether our functions are one-to-one is to sketch their curves.

Letβs start with a graph of π₯ to the fourth power plus π₯ squared. This is sometimes called a quartic graph. It has a positive leading coefficient of π₯. And if weβre to factor the expression, we see that it only has one root, and thatβs at π₯ equals zero. And so, the graph of π¦ equals π₯ to the fourth power plus π₯ squared looks a little something like this. It does indeed pass the vertical line test, a vertical line intersects the curve exactly once. However, if we add a horizontal line here, we see that it intersects the curve twice. And this means our function is not one-to-one. In this case, an element of the range can correspond to more than one element of its domain.

Now, in fact, our graph of π of π₯ equals π₯ squared looks very similar. And so, by the same reasoning, it cannot be one-to-one. Weβll now move on to the graph of π of π₯ equals cos of π₯. We know one full period of the graph of π¦ equals cos of π₯ looks a little something like this. Once again, this craft does indeed pass the vertical line test. But it absolutely doesnβt pass the horizontal line test. And since the graph of π¦ equals cos of π₯ is periodic, we can see that a horizontal line will intercept the curve of π¦ equals cos of π₯ an infinite number of times. And so, π of π₯ equals cos of π₯ cannot be a one-to-one function.

Weβll now test π of π₯ equals π₯ cubed. The graph of π¦ equals π₯ cubed looks like this. And we can see it quite clearly passes the vertical line test. This time it also passes the horizontal line test. A horizontal line added to the graph intersects that curve exactly once. So, we can say that the correct answer is d. The one-to-one function is π of π₯ equals π₯ cubed.