Question Video: Identifying One-to-One Functions from their Graph | Nagwa Question Video: Identifying One-to-One Functions from their Graph | Nagwa

Question Video: Identifying One-to-One Functions from their Graph Mathematics • Second Year of Secondary School

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

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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.

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