Which of the following is a one-to-one function? Is it a) 𝑓 of 𝑥 equals 𝑥 to the fourth power plus 𝑥 squared, b) 𝑓 of 𝑥 equals 𝑥 squared, c) 𝑓 of 𝑥 equals cos of 𝑥, or d) 𝑓 of 𝑥 equals 𝑥 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.
Let’s start with a graph of 𝑥 to the fourth power plus 𝑥 squared. This is sometimes called a quartic graph. It has a positive leading coefficient of 𝑥. And if we’re to factor the expression, we see that it only has one root, and that’s at 𝑥 equals zero. And so, the graph of 𝑦 equals 𝑥 to the fourth power plus 𝑥 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 𝑓 of 𝑥 equals 𝑥 squared looks very similar. And so, by the same reasoning, it cannot be one-to-one. We’ll now move on to the graph of 𝑓 of 𝑥 equals cos of 𝑥. We know one full period of the graph of 𝑦 equals cos of 𝑥 looks a little something like this. Once again, this craft does indeed pass the vertical line test. But it absolutely doesn’t pass the horizontal line test. And since the graph of 𝑦 equals cos of 𝑥 is periodic, we can see that a horizontal line will intercept the curve of 𝑦 equals cos of 𝑥 an infinite number of times. And so, 𝑓 of 𝑥 equals cos of 𝑥 cannot be a one-to-one function.
We’ll now test 𝑓 of 𝑥 equals 𝑥 cubed. The graph of 𝑦 equals 𝑥 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 𝑓 of 𝑥 equals 𝑥 cubed.