Schaum's Outline of Theory and Problems of Beginning Calculus

~~~~~~~~ ~ ~~~ ~ ~ ~ ~ ~ ~ ~Chapter 3737.1 ONEONE FUNCTIONSInverse Trigonometric FunctionsIn Section 35.3, we introduced the notion of the inverse of a function, and we showed that theinverse of In x is ex, and vice versa. Not all functions, however, have inverses.EXAMPLES(a) Consider the function f such that f(x) = x2 for all x. Then f( 1) = 1 and f( - 1) = 1. If there were an inverse gof f, then g(f(x)) = x. Therefore, g(1) = g(f(1)) = 1 and g(1) = g(f(- 1)) = - 1, implying that 1 = - 1,which is impossible.(b) Letfbe any periodic function,f(x + p) =f(x), for all x (see Section 26.2). The argument of example (a), for twopoints xo and xo + p, shows that f cannot have an inverse. Now all the trigonometric functions are periodic(with either p = 2n or p = n). Hence, the trigonometric functions do not have inverses!The functions that have inverses turn out to be the one-one functions.Definition: A functionfis one-one if, whenever U # v,f (U) # f (0)Thus, a one-one function takes different numbers into different numbers. A function is one-one ifand only if its graph intersects any horizontal line in at most one point. Figure 37-l(a) is the graph of aone-one function; Fig. 37-l(b) graphs a function that is not one-one because f (U) =f(t)) = c.-7*XYc-J1 1 *UUX*Fig. 37-1NOTATION The inverse of a one-one function f will be denoted by f -fWarning: Do not confuse the inversef-' with the reciprocal 16292