WEBVTT
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Which of the following is a one-to-one function?
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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?
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We begin by recalling what it actually means for a function to be one-to-one.
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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.
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One way of testing for a one-to-one function is to consider the shape of the graph.
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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.
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Then, we can say a function is one-to-one.
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And so, the easiest way to check whether our functions are one-to-one is to sketch their curves.
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Letβs start with a graph of π₯ to the fourth power plus π₯ squared.
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This is sometimes called a quartic graph.
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It has a positive leading coefficient of π₯.
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And if weβre to factor the expression, we see that it only has one root, and thatβs at π₯ equals zero.
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And so, the graph of π¦ equals π₯ to the fourth power plus π₯ squared looks a little something like this.
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It does indeed pass the vertical line test, a vertical line intersects the curve exactly once.
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However, if we add a horizontal line here, we see that it intersects the curve twice.
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And this means our function is not one-to-one.
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In this case, an element of the range can correspond to more than one element of its domain.
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Now, in fact, our graph of π of π₯ equals π₯ squared looks very similar.
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And so, by the same reasoning, it cannot be one-to-one.
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Weβll now move on to the graph of π of π₯ equals cos of π₯.
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We know one full period of the graph of π¦ equals cos of π₯ looks a little something like this.
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Once again, this craft does indeed pass the vertical line test.
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But it absolutely doesnβt pass the horizontal line test.
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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.
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And so, π of π₯ equals cos of π₯ cannot be a one-to-one function.
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Weβll now test π of π₯ equals π₯ cubed.
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The graph of π¦ equals π₯ cubed looks like this.
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And we can see it quite clearly passes the vertical line test.
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This time it also passes the horizontal line test.
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A horizontal line added to the graph intersects that curve exactly once.
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So, we can say that the correct answer is d.
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The one-to-one function is π of π₯ equals π₯ cubed.