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# Algebra: Praxis I Exam (page 5)

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Updated on Jul 5, 2011

### 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.

### Elimination

The elimination method involves writing one equation over another and then adding or subtracting the like terms on the same sides of the equal sign so that one letter is eliminated.

Example
x – 9 = 2y and x – 3 = 5y
• Rewrite each equation in the same form.
x – 9 = 2y becomes x – 2y = 9 and x – 3 = 5y becomes x – 5y = 3.
• If you subtract the two equations, the "x" terms will be eliminated, leaving only one variable: Subtract:
x – 2y = 9
–(x – 5y = 3)
• y = 2 is the answer.
• Substitute 2 for y in one of the original equations and solve for x.
x – 9 = 2y
x – 9 = 2(2)
x – 9 = 4
• x – 9 + 9 = 4 + 9
x = 13
• The answer to the system of equations is y = 2 and x = 13.

If the variables do not have the same or opposite coefficients as in the preceding example, adding or subtracting will not eliminate a variable. In this situation, it is first necessary to multiply one or both of the equations by some constant or constants so that the coefficients of one of the variables are the same or opposite. There are many different ways you can choose to do this.

Example
3x + y = 13
x + 6y = –7

We need to multiply one or both of the equations by some constant that will give equal or opposite coefficients of one of the variable. One way to do this is to multiply every term in the second equation by –3.

3x + y = 13
–3(x + 6y = –7) –3x – 18y = 21

Now if you add the two equations, the "x" terms will be eliminated, leaving only one variable. Continue as in the preceding example.

3x + y = 13
–3x – 18y = 21
y = –2 is the answer.
• Substitute –2 for y in one of the original equations and solve for x.
3x + y = 13
3x + (–2) = 13
3x + (–2) + –2= 13 + –2
• 3x = 11
• The answer to the system of equations is y = –2 and x = .

### Inequalities

Linear inequalities are solved in much the same way as simple equations. The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction.

Example

 10 > 5 so (10)(–3) < (5)(–3) –30 < –15

### Solving Linear Inequalities

To solve a linear inequality, isolate the letter and solve the same way as you would in a linear equation. Remember to reverse the direction of the inequality sign if you divide or multiply both sides of the equation by a negative number.

Example
If 7 – 2x > 21, find x.
• Isolate the variable.
7 – 2x > 21
• Because you are dividing by a negative number, the inequality symbol changes direction.
x < –7
• The answer consists of all real numbers less than –7.

### Solving Compound Inequalities

To solve an inequality that has the form c < ax + b < d, isolate the letter by performing the same operation on each part of the equation.

Example
If –10 < –5y – 5 < 15, find y.
• Add five to each member of the inequality.
–10 + 5 < –5y – 5 + 5 < 15 + 5 – 5 < –5y < 20
• Divide each term by –5, changing the direction of both inequality symbols:
• The solution consists of all real numbers less than 1 and greater than –4.