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# Algebra: Praxis I Exam (page 4)

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Updated on Jul 5, 2011

### Patterns and Functions

The ability to detect patterns in numbers is a very important mathematical skill. Patterns exist everywhere in nature, business, and finance.

When you are asked to find a pattern in a series of numbers, look to see if there is some common number you can add, subtract, multiply, or divide each number in the pattern by to give you the next number in the series.

For example, in the sequence 5, 8, 11, 14…you can add 3 to each number in the sequence to get the next number in the sequence. The next number in the sequence is 17.

Example
What is the next number in the sequence , 3, 12, 48?
Each number in the sequence can be multiplied by the number 4 to get the next number in the sequence: × 4 = 3, 3 × 4 = 12, 12 × 4 = 48, so the next number in the sequence is 48 × 4 = 192.

Sometimes it is not that simple. You may need to look for a combination of multiplying and adding, dividing and subtracting, or some combination of other operations.

Example
What is the next number in the sequence 0, 1, 2, 5, 26?
Keep trying various operations until you find one that works. In this case, the correct procedure is to square the term and add 1: 02 + 1 = 1, 12 + 1 = 2, 22 + 1 = 5, 52 + 1 = 26, so the next number in the sequence is 262 + 1 = 677.

### Properties of Functions

A function is a relationship between two variables x and y where for each value of x, there is one and only one value of y. Functions can be represented in four ways:

1. a table or chart
2. an equation
3. a word problem
4. a graph

For example, the following four representations are equivalent to the same function:

Helpful hints for determining if a relationship is a function:
• If you can isolate y in terms of x using only one equation, it is a function.
• If the equation contains y2, it will not be a function.
• If you can draw a vertical line anywhere on a graph such that it touches the graph in more than one place, it is not a function.
• If there is a value for x that has more than one y-value assigned to it, it is not a function.

Example

 x = 5 Contains no variable y, so you cannot isolate y. This is not a function. 2x + 3y = 5 x2 + y2 = 36 Contains y2, so it is not a function. |y| = 5 There is no way to isolate y with a single equation; therefore, it is not a function.

### Function Notation

Instead of using the variable y, often you will see the variable f(x). This is shorthand for "function of x" to automatically indicate that an equation is a function. This can be confusing; f(x) does not indicate two variables f and x multiplied together; it is a notation that means the single variable y.

Although it may seem that f(x) is not an efficient shorthand (it has more characters than y), it is very eloquent way to indicate that you are being given expressions to evaluate. For example, if you are given the equation f(x) = 5x – 2, and you are being asked to determine the value of the equation at x = 2, you need to write "evaluate the equation f(x) = 5x – 2 when x = 2." This is very wordy. With function notation, you only need to write "determine f(2)." The x in f(x) is replaced with a 2, indicating that the value of x is 2. This means that f(2) = 5(2) – 2 = 10 – 2 = 8.

All you need to do when given an equation f(x) and told to evaluate f(value) is to replace the value for every occurrence of x in the equation.

Example
Given the equation f(x) = 2x2 + 3x + 1, determine f(0) and f(–1).
f(0) means replace the value 0 for every occurrence of x in the equation and evaluate.
f(0) = 2(0)2 + 3(0) + 1
= 0 + 0 + 1
= 1
f(–1) means replace the value –1 for every occurrence of x in the equation and evaluate.
f(0) = 2(–1)2 + 3(–1) + 1
= 2(1) + –3 + 1
= 2 – 3 + 1
= 0

### Families of Functions

There are a number of different types, or families, of functions. Each function family has a certain equation and its graph takes on a certain appearance. You can tell what type of function an equation is by just looking at the equation or its graph.

These are the shapes that various functions have. They can appear thinner or wider, higher or lower, or upside down.

### Systems of Equations

A system of equations is a set of two or more equations with the same solution. Two methods for solving a system of equations are substitution and elimination.

### Substitution

Substitution involves solving for one variable in terms of another and then substituting that expression into the second equation.

Example
2p + q = 11 and p + 2q = 13
• First, choose an equation and rewrite it, isolating one variable in terms of the other. It does not matter which variable you choose.
2p + q = 11 becomes q = 11 – 2p.
• Second, substitute 11 – 2p for q in the other equation and solve:
p + 2(11 – 2p) = 13
p + 22 – 4p = 13
22 – 3p = 13
22 = 13 + 3p
9 = 3p
p = 3
• Now substitute this answer into either original equation for p to find q.
2p + q = 11
2(3) + q = 11
6 + q = 11
• q = 5
• Thus, p = 3 and q = 5.