Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 371 INVERSE TRIGONOMETRIC FUNCTIONS 293If a one-one function f is defined by means of a formula, y =f(x), we can sometimes solve thisequation for x in terms of y. This solution constitutes the formula for the inverse function, x =f- '(y).EXAMPLES(a) Letf(x) = 3x + 1 (a one-one function). Let y stand forf(x); then y = 3x + 1. Solve this equation for x in termsof Y,Therefore, the inversef-' is given by the formula(6) Consider the one-one functionf(x) = 2e" - 5,Thus,y-l=3x-- Y-Lx3Y-1 f-'(y) = - 3y = 2e" - 5y + 5 = 2e"-- Y+5- ex2Y+5In - = In ex = x2(c)Let f(x) = xs + x. Since f'(x) = 5x4 + 1 > 0, f is an increasing function, and therefore one-one (see Problem37.12). But if we write y = xs + x, we have no obvious way of solving the equation for x in terms of y.37.2 INVERSES OF RESTRICTED TRIGONOMETRIC FUNCTIONSFor a periodic function to become one-one-and so to have an inverse-its domain has to berestricted to some subset of one period.Inverse SineThe domain off(x) = sin x is restricted to [ - 42, 421, on which the function is one-one [in fact,increasing; see Fig. 37-2(a)]. The inverse function f - '(x) = sin - ' x is called the inverse sine of x (or,sometimes, the arc sine of x, written arc sin x or arcsin x). Its domain is [ - 1, 13. Thus,tY(a) y = sin x(b) y = sin-'xFig. 37-2