### Video Transcript

Which curve among those shown in
the graph below is a one-to-one function?

In order to answer this question,
we recall the horizontal line test, which states that a function is one-to-one or
injective if and only if its graph intersects with each horizontal line at most
once. Let’s begin with the red graph. Our goal is to draw, if possible, a
horizontal line that crosses the graph more than once, thereby indicating that the
function is not injective. One way of doing this would be to
draw the horizontal line with equation 𝑦 is equal to negative 10. We see that this horizontal line
crosses the red graph twice. We can therefore conclude that the
function represented by the red graph is not one-to-one.

We can repeat this process for the
green graph by drawing a horizontal line with equation 𝑦 equals 10. This horizontal line crosses the
green graph three times. We can therefore also conclude that
the function represented by the green graph is not one-to-one. We can also repeat this process for
the yellow graph by drawing a horizontal line with equation 𝑦 is equal to two. This horizontal line crosses the
yellow graph twice. So by the horizontal line test, the
function represented by the yellow graph is not one-to-one. As we have proved that the red,
green, and yellow graphs do not represent one-to-one functions, this suggests that
the blue graph does represent a one-to-one function. By firstly clearing the other
horizontal lines we have drawn, let’s consider this function in isolation.

We begin by drawing several
horizontal lines, in this case, at 𝑦 equals 20, 30, 40, and 50. Each of these horizontal lines
intersects our blue graph once and only once. We can check that this is true for
the whole graph by sliding a horizontal ruler up and down the graph. Each horizontal line that could be
drawn on the graph crosses the blue graph at most once. We can therefore conclude by the
horizontal line test that the function represented by the blue graph is one-to-one
or injective. This is the only one of the four
curves that represents a one-to-one function.