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Which of the following graphs represents a function π of π₯ and its inverse function the inverse π of π₯?
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Letβs begin by reminding ourselves what it means for us to be talking about the inverse function.
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Suppose we have some function π.
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The inverse function, represented by the superscript negative one, undoes that original function.
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So the inverse function of π of π₯ is equal to π₯ for all values of π₯ in the domain of the function.
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But what does this actually mean if we think about the graphical representation of these functions?
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Well, if we plot π¦ equals π of π₯ on the coordinate plane, we can find the graph of the inverse function π¦ is equal to the inverse π of π₯ by reflecting the original graph across the line π¦ equals π₯.
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So letβs draw the line π¦ equals π₯ on each of our graphs and see if we can identify which one represents π of π₯ and its inverse function.
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The line π¦ equals π₯ passes through the origin, the point zero, zero, and it has a slope of one, as shown.
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Now, in fact, we can see that only one of these graphs represents a pair of functions which have been reflected across the line π¦ equals π₯.
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Specifically, we see that this is graph one.
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We can check this by identifying a couple of points that lie on each of these graphs.
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For instance, it appears as if the point negative 7.50 lies on the blue plot in graph one.
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Reflecting this point across the line π¦ equals π₯ essentially has the result of switching the π₯- and π¦-values.
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And we do indeed see that we get the point with coordinates zero, negative 7.5.
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So the graph that represents a function π of π₯ and its inverse function is graph one.
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Now, we note it doesnβt actually matter which of these plots represents the inverse function and which of the plots represents the original function.
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The function and its inverse function will always be inverses of one another.