# Section 15.5: Functions

## 15.5 Outline

1. Definitions
1. first/second components
2. function
2. Representations of functions
1. ordered pairs
2. mapping
3. machine
3. Functional notation
1. domain
2. range
3. difference quotient
4. Vertical line test
5. Some important functions (see below)
1. constant function
2. linear function
4. trigonometric function
5. exponential function
6. logarithmic function
7. probability function

## 15.5 Essential Ideas

A function is a set of ordered pairs in which the first component is associated with exactly one second component. Use the vertical line test to see if a given graph is a function.

Here are some important functions:

constant function:

f (x ) = k

linear function:

f (x ) = mx + b (m is not zero)

f (x ) = ax2 + bx + c    (a is not zero)

exponential function:

f (x ) = bx    (b> 0, b is not one)

logarithmic function:

f (x ) = log bx   (b> 0, b is not zero, x> 0)

probability function: A function P which satisfies the following properties:

0 ≤ P (E ) ≤ 1 P (S ) = 1      S is the sample space)

P (E or F ) =P (E ) +P (F )     (E and F are mutually exclusive)