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 Dependent Variable

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# Determining if a Function has an Inverse

Some functions do not have an inverse function. For example, letâ€™s consider f(x) = x2. Below is a table of values and the graph for f(x) = x2.

In the table, each input corresponds to exactly one output. We also can see that the graph passes the vertical line test since any vertical line intersects the graph at most once. Thus, f(x) = x2 is a function.

If we attempt to form the inverse function for f(x) = x2 by switching the x and y values, the following table and graph result.

This table does not represent a function. We can see this because there are values of x that correspond to two values of y.

For example, when x = 4, y = both 2 and -2. Thus, the graph does not pass the vertical line test since a vertical line may intersect the graph more than once.

Since the second table does not represent a function, f(x) = x2 does not have an inverse function.