1. §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

2. 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.

5. (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

7. (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.

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

12. 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 = (– ,)

26. (7) Cancellation:

33. 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

35. (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

37. (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

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