CHAPTER 1

A function f is a rule that assigns to each element x in a set A exactly one element, called f (x), in a set B.

The set A is called the domain of the function.

The range of the function is the set of all possible values of f (x) as x varies throughout the domain.

Increasing and Decreasing Functions

A function f f is called increasing on an interval I if

f(x1) < f(x2) whenever x1 < x2 in I.

Increasing and Decreasing Functions

A function f f is called decreasing on an interval I if

f(x1) > f(x2) whenever x1 < x2 in I.

1.5 Exponential Functions

An exponential function is afunction is a function of the form f(x) = ax where a is a positive constant.If x = n, a is a positive integer, then an = a .a . a… . … . a (n factors)

If x = 0,then a0=1, and if x = -nwhere n is a positive integer, then a-n=1/an .If x is a rational number, x= p/qwhere p and q are integers and q > 0, then ax= a p/q = qa p .

Laws of Exponents If a and bare positive numbers and x and y are any real numbers, then1. ax+y = ax ay

2. ax-y = ax /ay 3. (ax) y = ax y 4. (ab)x = axby

Definition of E In the family ofexponential functions f(x) = bx

there is exactly one exponentialFunction for which the slope of theLine tangent at (0,1) is exaclty 1.This occurs for b=2.71…This Important number is denoted by e.

1.6 Inverse Functions and Logarithms

Definition A function f is called a one-to-one function if it nevertakes on the same value twice; that is,

f(x1) is not equal to f(x2)whenever x1 is not equal to x2 .

Horizontal Line Test A function isone-to-one if and only if no horizontal line intersects its graph more than once.

1.6 Inverse Functions and Logarithms

Definition Let f be a one-to-one function with domainA and range B. Then its inversefunction f –1 has domain B and range A and is defined by f –1(y) = x, then f(x) = yfor any y in B.

domain of f –1 = range of f range of f –1 = domain of f . f –1 (x) = y then f(y) = x. f –1 (f (x)) = x for every x in A f (f –1(x)) = x for every x in B.

The graph of f –1 is obtained by reflecting the graph of f about the line y = x.

The graph of f –1 is obtained by reflecting the graph of f about the line y = x.

ln x = y then e y = x

The graph of f –1 is obtained by reflecting the graph of f about the line y = x.

Laws of LogarithmsIf x and y are positive numbers, then• ln (x y) = ln x + ln y• ln (x/y) = ln x - ln y• ln (xr) = r ln x (where r is any real number)

ln x = y then e y = x

ln(e x) = x x R e ln x = x x > 0

ln x = y then e y = x

ln x = y then e y = x

ln e = 1

ln(e x) = x x R e ln x = x x > 0

For any positive number a ( a is not equal to 1), we have log a x = ln x / ln a