A function maps an input value x into an output value y = f(x). There are cases in which we want to go the other way around; i.e., given y, we would like to find a value x such that y = f(x). Actually, this is what we do whenever we want to solve an equation. For instance, given a function that evaluates the NPV of an investment depending on the discount rate r, finding its IRR calls for the solution of the equation NPV(r) = 0. Solving an equation like that requires inversion of the mapping associated with the function for one specific value of NPV, zero in this case. Now imagine that we want to do something more. We would like to find a trick that allows us to solve an entire range of equations f(x) = y for many values of y. This idea leads to the definition of the inverse function of f, mapping y to the corresponding x. As you may imagine, this cannot be done for every function f, as equations may have multiple solutions or none at all, whereas an inverse function should map one input value into one output value.
DEFINITION 2.5 (Inverse function) Given a function 
, where E1 is the domain of f, the inverse function of f is a function 
such that
- g(f(x)) = x, for all x in the domain E1 of f
- f(g(z)) = z, for all z in the domain E2 of g
Typically, the notation f−1 is used to denote the inverse of function x. This should not be confused with the composite function g(x) = 1/f(x). Some care in notation may avoid the confusion: f−1(x) refers to the inverse function, whereas [f(x)]−1 = 1/f(x) refers to the composite function.
Example 2.12 Consider function f(x) = (x − 1)/(x + 1). To find its inverse, we set up and solve the following equation:
Hence, the inverse function of f is
We see that the inverse function is not defined for y = 1. Indeed
which is absurd. The image of function f does not include the value 1, and the inverse function is not defined there.
Fig. 2.14 Square root as an inverse function on a restricted domain.
Example 2.13 Consider function f(x) = x2. It may be tempting to say that its inverse is the square-root function 
, but a quick look at plots in Fig. 2.14 points out a difficulty. An equation like x2 = 4 has two roots, x = ±2; which one should we use when inverting the function? As shown in the figure, inverting the function essentially means swapping the two coordinate axes, but in doing so we do not always define a true function, which should map each value of the independent variable into one value. In this case, the domain of f(x) = x2 must be restricted in order to define its inverse function. The customary choice is restricting the domain to positive numbers, so we just consider the positive root. In other words, we cancel the dashed lower part of the rotated parabola in the second plot of Fig. 2.14.
The last example shows that not every function can be inverted over its whole domain. In order to be invertible, f should not assign the same value to two different arguments:
A function whose graph goes up and down does not have this property. Another unpleasing feature that may prevent inversion is lack of continuity. If a function jumps, then we may fail to invert it. A condition ensuring invertibility of a function is that it is continuous and strictly increasing:
If this condition is met on an interval, then the function is invertible on that interval. It is also easy to see this also applies to a continuous strictly decreasing function.
Fig. 2.15 The natural logarithm.
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