§1.5 Inverse and log functions In this section, we look at logarithmic functions, which are another very important type of functions. We define logarithmic functions as inverse of exponential functions. Topics: Inverse functions Logarithmic functions Inverse trigonometric functions

Inverse Functions (1) One-to-one functions Definition: A function f is called a one-to-one function if x1  x2 implies f(x1)  f(x2). Comment: A function may send many elements in its domain to one element in its range. So a one-to-one function rules out many-to-one. Ex: y = f(x) = x2. We see f(–1) = f(1). So f is not a one-to-one function.

(3) The x is traditionally used as the independent variable, so when we focus on f –1, we usually write y = f –1(x). (4) How to find the defining rule of f –1 if f is 1-1? We follow these steps Write y = f(x) Solve x in terms of y Interchange x and y

(5) Graph of f –1 The graph of f –1 can obtained by reflecting the graph of f about the line y = x. Reason: If (a,b) is on the graph of f  f(a) = b  f –1 (b) = a  (b,a) is on the graph of f –1. (a,b) and (b,a) are symmetric about the line y = x.

(6) Cancellation: f –1(f(x)) = x , f(f –1(y)) = y

Logarithmic functions (1) Definition If a > 0 and a  1, then f(x) = ax is one-to-one by the horizontal line test. So it has an inverse function f –1, which is called the logarithmic function with base a and denoted by loga. Note: (i) loga x = y  ay = x (ii) Let f(x) = loga x, where a > 0 but a1. Then Df = (0,), Rf = (– ,)

(7) Cancellation:

C. Inverse trigonometric functions (1) Inverse of sin sin(x) is not 1-1. But f(x) = sin(x), – /2  x  /2 is 1-1. Its inverse function, denoted by sin –1 or arcsin, is called the inverse function of sin

(2) Inverse of cos g(x) = cos(x), 0  x   is 1-1. Its inverse function, denoted by cos-1 or arccos, is called the inverse function of cos

(3) Inverse of tan h(x) = tan(x), – /2 < x < /2 is 1 – 1. Its inverse function, denoted by tan-1 or arctan, is called the inverse function of tan

(4) Other inverse trig functions Likewise, we can define other trig functions.