1.9. Inverse Functions
There is a very important operation on one-to-one functions, known as the inverse. For a given one-to-one function f(x), the inverse of f, denoted f-1, is defined by x = f-1(y) whenever y = f(x). Although the inverse is technically only well-defined for one-to-one functions, useful inverses are defined for many other functions by restricting their domains to regions in which they are one-to-one.
Example 1: Find the inverse of the function f(x) = 2x.
Solution: From Section 1.7, we know that f(x) is one-to-one, so it should have a well-defined inverse. To find it, we note that y = f(x) = 2x if and only if x = f-1(y) = y/2. Thus, the inverse of f(x) is the function f-1(x) = x/2.
Example 2: Find the inverse of the function f(x) = x2.
Solution: From Section 1.7, we know that f(x) is not one-to-one, so it does not have a well-defined inverse. However, we may still find a useful inverse of f if we restrict its domain appropriately. If we restrict the domain of f to the nonnegative real numbers, then we see that f is now one-to-one. What is the inverse of f restricted to this domain? Note that y = f(x)=x2 implies that x = f-1(y) = ±√y. Which sign should we take? Since we have restricted the domain of f to the postive real numbers, we see that we need to take the positive square root. Thus, f-1(x) = √x is the inverse of f(x)=x2. (More generally, the the kth root function f-1(x) = k√x is the inverse of the kth power function f(x)=xk.)
There is a nice geometric interpretation of the inverse of a function. Since y=f(x) implies x=f-1(y), we see that interchanging the roles of x and y transforms a function to its inverse. But interchanging x and y merely reflects the graph of y=f(x) about the diagonal line y=x. Thus, the graph of the inverse of a function is a reflection of the graph of the original function about the line y=x. Functions whose graphs are symmetric about this line, such as f(x)=x, f(x)=-x, and f(x)=1/x, are equal to their own inverse.