inverse function
Mathematical function that undoes the effect of another function. For example, the inverse function of the formula that converts Celsius temperature to Fahrenheit temperature is the formula that converts Fahrenheit to Celsius. Applying one formula and then the other yields the original temperature. Inverse procedures are essential to solving equations because they allow mathematical operations to be reversed (e.g. logarithms, the inverses of exponential functions, are used to solve exponential equations). Whenever a mathematical procedure is introduced, one of the most important questions is how to invert it. Thus, for example, the trigonometric functions gave rise to the inverse trigonometric functions.
inverse function [′in‚vərs ′fənk·shən]
Inverse Function
a function that is the inverse of a given function. For example, if y = f(x) is a given function, then the variable x, considered as a function of the variable y, x = ø(y), is the inverse of the function y = f(x). For example, the inverse function of y = ax + b (α ≢ 0) is x = (y ø b)/a, the inverse function of y = ex is x = 1n y, and so forth. If x = ø(y) is the inverse function of y = f(x), then y = f(x) is the inverse function of x = ø(y).
The domain of definition of the given function is the range of values of the inverse function, and the range of the given function is the domain of the inverse function. The graphs of two mutually inverse functions y = f(x) and y = ø(x) (where the independent variable is designated by the same letter x), such as y = ax + b and y = (x – b)/a or y = ex and y = 1n x, are symmetric with respect to the line y = x, which bisects the first and third quarters of the coordinate plane. The inverse of a single-valued function may be multiple-valued; compare, for example, the functions x2 and 
. A function y = f(x) has a single-valued inverse if and only if it takes on different values for different values of the independent variable. For a continuous function, this last condition can be satisfied only if the given function is monotonic (we have in mind real-valued functions of a real variable). The inverse of a continuous and monotonic function is single-valued, continuous, and monotonic.
If a given function is piecewise monotonic, then division of its domain of definition into regions of monotonicity yields single-valued branches of its inverse. For example, the interval –π/2 < x < π/2 serves as one of the regions of monotonicity of sin x. The branch of the inverse function Arc sin x that corresponds to this interval is the so-called principal branch arc sin x. The relations ø[f(x)] = x and f[ø(x)] = x hold for a pair of single-valued mutually inverse functions. The former holds for all values of x in the domain of definition of f(x), and the latter for all values of x in the domain of definition of ø(x); for example, e1nx = x(x > 0) and ln (ex) = x (— ∞ < x < ∞). The inverse of f(x) = y is sometimes designated as f1(y) = x, so that for a continuous and monotonie function f(x),
f–1[f(x)] = f[f–1f(x)] = x
In general, f–1[f(x)] is a multiple-valued function of x, one of whose values is x; for example, for f(x) = x2, x (≠ 0) is only one of the two values of 
(the other is – x); for f(x) = sin x, x is only one of an infinite set of values
f–1[f(x)] = Arc sin [sin x] = (–1)nx + nπ,
n = 0, ±1, ±2, …
If y = f(x) is continuous and monotonie in a neighborhood of x = x0 and has a nonzero derivative f’(x0) at x = x0, then f–1(y) is differentiable at y = y0, and
[f–1(y0)]’ = 1/f’(x0)
(the differentiation formula for an inverse function). Thus, for –π/2 < x < π/2,y = f(x) = sin x is continuous and monotonic and f’(x) = cos x ≠ 0. Then f–1(y) = arc sin y (–1 < y < 1) is differentiable, and
Here the root is positive since cos x > 0 for –π/2 < x < π/2.
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