Inverse Functions - Inverses in Calculus

Inverses in Calculus

Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functions are often defined through formulas, such as:

A function ƒ from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as long as the graph of y = ƒ(x) has, for each possible y value only one corresponding x value, and thus passes the horizontal line test.

The following table shows several standard functions and their inverses:

Function ƒ(x)	Inverse ƒ−1(y)	Notes
x + a	y – a
a – x	a – y
mx	y / m	m ≠ 0
1 / x	1 / y	x, y ≠ 0
x2		x, y ≥ 0 only
x3		no restriction on x and y
xp	y1/p (i.e. )	x, y ≥ 0 in general, p ≠ 0
ex	ln y	y > 0
ax	loga y	y > 0 and a > 0
trigonometric functions	inverse trigonometric functions	various restrictions (see table below)