Algebra and Functions
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
You may need to apply these types of factoring:
x2 + 2x = x (x + 2)
x2 – 1 = (x + 1) (x – 1)
x2 + 2x + 1 = (x + 1) (x + 1) = (x + 1)2
2x2 + 5x – 3 = (2x – 1) (x + 3)
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.
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 .
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.
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: