Math Concepts

Algebra and Functions

Register Next Tests:
5/3 , 6/7

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) =  square root (x + 2), 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) = square root (14 + 2) = square root 16 = 4.

Note: the square rootsymbol represents the positive, or principal, square root. For example, square root 16 = 4, not ±4.


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

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

(x^a) times (x^b) = x^(a + b)    (x^a)^b = x^(a times b)    (x times y)^a = (x^a) times (y^a)

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

x^a over x^b = x^(a minus b)    (x over y)^a = x^a over y^a    x^negative a = 1 over (x^a)

Also, x^(a over b) = b root (x^a) . For example,  x^(2 over 3) = cube root (x^2).

Note: For any nonzero number x, it is true that x^0 = 1.


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 y = k over x 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

absolute value = x comma if x is greater than or equal to 0 or negative x comma if x is less than 0

for example colon absolute value 2 = 2 comma since 2 is greater than 0 and absolute value negative 2 = negative (negative 2) = 2 comma since negative 2 is less than 0 and absolute value 0 = 0

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