Inverse Functions

Two real functions f and g are called inverse functions if the two equations

y = f(x), x = g(y)

have the same graphs in the (x, y) plane. That is, a point (x, y) is on the curve y = f (x) if, and only if, it is on the curve x = g(y). (In general, the graph of the equation x = g(y) is different from the graph of y = g(x), but is the same as the graph of y = f(x); see Figure 2.4.1.)

For example, the function y = x², x ≥ 0, has the inverse function x = √y; the function y = x³ has the inverse function x=

If we think of f as a black box operating on an input x to produce an output f(x), the inverse function g is a black box operating on the output f(x) to undo the work of f and produce the original input x (see Figure 2.4.2).