The Range of the Inverse

The last two post discussed inverse functions and some concerns about them. We continue that today be considering that fact that sometimes the inverse of a function is not a function, and what can be done in that case.

Since the square of both 3 and –3 is 9. Which number should you get when you unsquare 9? Is the result, 3 or –3?

Mathematicians want, for practical reasons, inverses to be functions. If the original function is not strictly monotonic then the inverse will not be a function. That is, if there are places on the original function that have the same y-values then the inverse (set of ordered pairs found by reversing the function’s ordered pairs) will not be a function. If some horizontal line intersects the graph of the function more than once, the inverse will not be a function.

While it may seem a bit too convenient, what is done is that the range of the inverse is restricted so that the inverse is a function. So for f (x) = x2 the range of the inverse is restricted to non-negative values. So f -1(9) = 3 and f -1(10) = \sqrt{10}  where it is understood that this represents a non-negative number. This is why {{f}^{-1}}\left( {{a}^{2}} \right)=\sqrt{{{a}^{2}}}=\left| a \right|. So that if a = –4, \sqrt{{{a}^{2}}}=\left| -4 \right|=4.

The restriction is arbitrary. It would be just as possible to make the range all non-positive numbers. While arbitrary, the restriction is not unreasonable. After all, once we understand this, we can easily find the other value if we need it. This is also necessary for calculators to work; the process they use to compute the value can only return one value. The (restricted) ranges of the common functions are what mathematicians feel are the most useful.

None of the trigonometric functions pass the horizontal line test; none of their inverses are functions until the ranges have been restricted. These restrictions are in the textbooks. For example: The domain of sin-1(x) is -1\le x\le 1, these are the output values of the sin(x); the range is restricted to \displaystyle -\tfrac{\pi }{2}\le {{\sin }^{-1}}\left( x \right)\le \tfrac{\pi }{2}. Because the signs of the trig functions are different outside of the first quadrant and in order to make as many of the inverses as possible continuous, each inverse trig function has a different range. You will find these in your textbook. They are built into calculators and computers. This can be a little confusing for students, but there is not much that can be done about that.

This is the third of 5 posts on inverses. The next post: The Calculus of Inverses.


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