The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Theorem 1: Inverse Function


f has an inverse function if and only if f is one-to-one.


The following statements are equivalent.

(1)    f is a one-to-one function.

(2)    For every y, either there is exactly one x with f(x) = y or there is no x with f(x) = y.

(3)    The equation y = f(x) determines x as a function of y.

(4)    f has an inverse function.


Every function which is increasing on its domain I has an inverse function. So does every function decreasing on its domain I.


Let f be increasing on I. For any two points x1 ≠ x2 in I, the value of f at the smaller of x1, x2 is less than the value of f at the greater, so f(x1) ≠ f(x2).

For example, the function y = x2 is not one-to-one because (-1)2 = 12, whence it has no inverse function. The function y = x2, x ≥ 0, is increasing on its domain [0, ∞) and thus has an inverse.

Now let us examine the trigonometric functions. The function y = sin x is not one-to-one. For example, sin 0 = 0, sin π = 0, sin 2π = 0, etc. We can see in Figure 7.3.3 that the inverse relation of y = sin x is not a function.


Figure 7.3.3

However, the function y = sin x is increasing on the interval [ - π/2, π/2], because its derivative cos x is ≥ 0. So the sine function restricted to the interval [-π/2, π/2],

y = sin x, - π/2 ≤ x ≤ π/2,

has an inverse function shown in Figure 7.3.4. This inverse is called the arcsine function. It is written x = arcsin y. Verbally, arcsin y is the angle x between - π/2 and π/2 whose sine is y.


Figure 7.3.4

Last Update: 2006-11-05