Function
A rule, or formula, which takes an input (or given starting value) and produces an output (or resulting value). For example, f(x) = x + 3 represents a function, where x is the input, f(x) is read as "f as a function of x" or "f of x" and refers to the output (also known as the "y" value), and x + 3 is the rule for what to do to the x input. f(4) = x + 3 = 4 + 3 = 7.
Independent Variable
The input variable of a function. In the function f(x) = x + 3, x represents the independent variable.
Dependent Variable
The output variable of a function. In the function f(x) = x + 3, f(x) represents the dependent variable, while x by itself represents the independent variable. f(x) does not mean "f times x" and the letter f is not a variable; rather, it is read as "f as a function of x" or "f of x."
Domain
All of the possible inputs, or numbers that can be used for the independent variable, for a given function. In the function f(x) = x^2, the domain is all numbers.
Range
All of the possible outputs, or numbers that can be used for the dependent variable, for a given function. In the function f(x) = x^2, the range is f(x) >= 0.
Compound, or Composite, Functions
Two nested functions; solved from the inner parentheses out. For example, f(g(x)) is an example of a compound function and is read as "f of g of x." Given f(x) = x + 3 and g(x) = 2x, g(x) is substituted first: f(g(x)) = f(2x). Next, f(x) is substituted: f(2x) = 2x + 3.
Direct Proportionality
Two given quantities are said to be "directly proportional" if the two quantities always change by the same factor and in the same direction. For example, doubling the input causes the output to double as well. The standard formula is y = kx, where x is the input, y is the output, and k is the proportionality constant (or the factor by which the numbers change). This equation can also be written as (y/x) = k, which means that the ratio of y to x is always the same constant.
Indirect Proportionality
Two given quantities are said to be "indirectly proportional" if the two quantities change by reciprocal factors. For example, doubling the input causes the output to halve. Tripling the input cuts the output to one-third of its original value. The standard formula is y = (k/x), where x is the input, y is the output and k is the proportionality constant. This equation can also be written as yx = k, which means that the product of y and x is always the same constant.
Proportionality Constant
The constant by which proportional values, whether direct or indirect, (above) change.
Linear Growth or Decay
The standard formula is y = mx + b, where x is the input, y is the output, slope m is the constant rate at which the quantity grows or declines, and b is the value of the quantity at time zero. Linear growth refers to a quantity that grows at a constant rate; a positive constant is added for each given period of time. Linear decay refers to a quantity that shrinks at a constant rate; a negative constant is added for each given period of time.
Exponential Growth or Decay
The standard formula is y = k^(Rt), where t is the time, k is the value of the quantity at time zero, R is the ratio, or constant multiplier, and y is the output. Exponential growth refers to a quantity that grows at a constant ratio; the same constant is multiplied for each given period of time and this constant is greater than 1. Exponential decay refers to a quantity that shrinks at a constant ratio; the same constant is multiplied for each given period of time and this constant is between zero and one.
Maximums or Minimums
The maximum possible output or minimum possible output of a given function. For example, the minimum output for the function f(x) = x^2 is 0. The maximum output is infinity.
Quadratic Functions
Functions whose rule, or formula, is a quadratic equation. See definition of Function (above) or the section on Quadratic Equations for more information.
