Determining if a Function has an Inverse
Some functions do not have an inverse function. For example, letâ€™s
consider f(x) = x^{2}. Below is a table of values and the graph for f(x)
= x^{2}.
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) = x^{2} is a function.
If we attempt to form the inverse function for f(x) = x^{2} 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) = x^{2} does not
have an inverse function.
