# Tools for Solving Equations

**Review: ABS Inequalities**

Every inequality containing only one absolute value can be

split into a compound inequality .

|x| < a −a < x < a

and

|x| > a x > a or x < −a

Think about why < is an AND while > is an OR.

**Review: Graphical Inequalities **

It is easy to think about this in terms of a graph if you get

everything on one side and a 0 on the other. For example,

turn |7x − 10| ≤ 9 into |7x − 10| − 9 ≤ 0 and then graph

y = |7x − 10| − 9

**Review: Quadratic Inequalities **

To solve a quadratic inequality, first make one side equal to

zero .

Then graph the equation when you make the zero y. For

example x^{2} − 3x − 4 > 0

**Review: Solving with Key Numbers**

Between key numbers, the sign of an expression will not

change.

Using this fact, we can solve the inequality

**§3.1: Functions**

A Relation is any way to take an x and find a y.

A Function is a relation where each x produces one, and

only one, y.

The Domain of a function are all of the x’s that are allowed

to be used.

The Range of a function are all the possible y’s that could

come out.

**Finding the Domain**

When you are fining the domain of a function, there are 2

possible problems:

1. You can’t divide by zero .

2. You can’t take the square root of a negative number .

So the domain of f(x) = 1/x is R except 0.

The domain of is non-negative elements of R

**How to answer these Questions**

For example:

Clearly x − 4 ≠ 0 so x ≠ 4

You could answer with the English statement “The domain

is all real numbers with the exception of 4”

Alternatively you could write:

R − {4}

**How to find the Range**

To determine the range of a function, say y = x^{2}, solve for x.

For what y’s can you find x’s?

In this case, you can find x’s only for y’s bigger than or

equal to 0, so the range is all non-negative numbers in R .

We will learn an easier way to find the range in the next

section.

What is the range of?

**§3.2 The graph of a function**

The graph of a relation is just all of the plotted points (x, y)

where x is related to y.

The graph of a function is just all of the plotted points (x, y)

where f(x) = y.

**The Vertical Line Test **

If a graph represents a function, it will pass the vertical line

test.

**Using a Graph to find Domain, Range**

Simply scan across to find the values where the function is

not defined at x. This is the domain.

Then scan up to find the values where the function is not

defined at y. This is the range.

So the domain is [-4,5] and the range is [-2,3) with [4,5].

**Graphing a piece-wise defined Function**

Graph

**§3.3 Rates of Change**

The average rate of change between a and b for a function

f(x) is

**Approaching Real Rates of Change**

The average rate of change between x and x + h when h is

small will be very close to the actual rate of change at x.

Examples:

#9

#23(a and b)

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