# Math Concepts

## Algebra and Functions

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### The following concepts are covered on the test:

- Substitution and simplifying algebraic expressions
- Properties of exponents
- Algebraic word problems
- Solutions of linear equations and inequalities
- Systems of equations and inequalities
- Quadratic equations
- Rational and radical equations
- Equations of lines
- Absolute value
- Direct and inverse variation
- Concepts of algebraic functions
- Newly defined symbols based on commonly used operations

### Factoring

You may need to apply these types of factoring:

x^{2} + 2x = x (x + 2)

x^{2} – 1 = (x + 1) (x – 1)

x^{2} + 2x + 1 = (x + 1) (x + 1) = (x + 1)^{2}

2x^{2} + 5x – 3 = (2x – 1) (x + 3)

### Functions

A function is a relation in which each element of the domain is paired with *exactly* one element of the range. On the SAT, unless otherwise specified, the domain of any function ƒ is assumed to be the set of all real numbers *x *for which ƒ(x) is a real number.

For example, if ƒ(x) = , the domain of ƒ is all real numbers greater than or equal to –2. For this function, 14 is paired with 4, since ƒ(14) = = = 4.

**Note:** the symbol represents the positive, or principal, square root. For example, = 4, not ±4.

### Exponents

You should be familiar with the following rules for exponents on the SAT.

For all values of *a*, *b*, *x*, *y*:

For all values of *a*, *b*, *x* > 0, *y* > 0:

Also, . For example, .

**Note**: For any nonzero number *x*, it is true that .

### Variation

**Direct Variation**: The variable *y* is directly proportional to the variable *x* if there exists a nonzero constant *k* such that *y = kx*.

**Inverse Variation**: The variable *y* is inversely proportional to the variable *x* if there exists a nonzero constant *k* such that or *xy = k*.

### Absolute Value

The absolute value of *x* is defined as the distance from *x* to zero on the number line. The absolute value of *x* is written as |*x*|. For all real numbers *x*:

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