**Important:** Even though we are starting with
**
**1.3, you should read all of
Chapter 1. This

should be mostly review of topics you've seen in previous courses.

I. Some Function Basics

**Definitions (from
1.1)
**

A

f(x), in a set B.

The set A is called the

Representations of Functions (from 1.1)

There are four possible ways to represent a function:

1. Verbally: by a description in words

2. Numerically: by a table of values

3. Visually: by a graph, or arrow diagram, or "machine"

4. Algebraically: by an explicit formula

The Vertical Line Test (from 1.1)

A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects

the curve more than once. (A graphical way to test if you have a function.)

Vertical and Horizontal Shifts (Translations)

Suppose c > 0. To obtain the graph of

y = f(x) + c, shift the graph of y = f(x) a distance c units upward

y = f(x) - c, shift the graph of y = f(x) a distance c units downward

y = f(x - c), shift the graph of y = f(x) a distance c units to the right

y = f(x + c), shift the graph of y = f(x) a distance c units to the left

**Vertical and Horizontal Stretching and Reflecting**

Suppose c > 1. To obtain the graph of

y = cf(x), stretch the graph of y = f(x) vertically by a factor of c

y = (1/c)f(x), compress the graph of y = f(x) vertically by a factor of c

y = f(cx), compress the graph of y = f(x) horizontally by a factor of c

y = f(x/c), stretch the graph of y = f(x) horizontally by a factor of c

y = -f(x), reflect the graph of y = f(x) about the x-axis

y = f(-x), reflect the graph of y = f(x) about the y-axis

**Example 3** Given f(x) = sin(2π x), graph f(2x), f(x/2), 2f(x), -f(x), and f(-x).

**Example 4 **(from Stewart) Problem 4, page 46, parts (c) and (d).

**III. Combinations of Functions
Algebra of Functions**

Let f and g be functions with domains A and B. Then the functions f + g, f - g, fg, and

f/g are defined as follows:

**Example 5** (from Stewart) Example 6, page 43. If
and , find

the functions f + g, f - g, fg, and f/g. Also, what is the new domain in each
case?

**IV. Composition of Functions
Definition**

Given two functions f and g, the

of f and g) is defined by

where the domain of f o g is the set of all x in the domain of g such that g(x) is in the

domain of f.

) What is the domain in each case?

of f does f o g represent?