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The Function Today

Today in most college algebra classes the function is thought of either as a rule, or as a set of ordered pairs.

The idea of a function as a rule is usually expressed by some formula or mathmatical expression in terms of 2 or more variables. Suppose a formula expresses the variable y in terms of the variable x. Then we say that y is a function of x. For example, consider y = 3x + 1. For each value of x, this formula determines the corresponding value of y. Let x = 2, then y = 3(2) + 1 = 7. This means 7 is the value of the function at x = 2. Functions are usually denoted by a letter, typically f, written y = f(x) and read "y equals f of x." This notation means that the value of y is determined by the value of x using the function f. x is called the independent variable, and y the dependent variable since its value depends on the value of x. One definition of the function as a rule is: A function is a rule that assigns to each member in one set (the domain) exactly one member from another set (the range).

The x values for which a function, f, is defined are called the domain of f. By defined we mean, when you plug in a value of x, the function value, f(x) is a real number. For example, let f(x) = 2/(x-1). If x = 1 f(x) = 2/0, but this is an undefined expression since you cannot divide by zero. Since this is the only number that makes f undefined, the domain of f would be all real numbers except x = 1. Another example, let g(x) = square root of x If x = -4, g(x)=2i, and this is a complex number, not a real number, thus anthing less than 0 would not be defined in g(x), therefore the domain of f is all real numbers greater than or equal to 0. The y values a function, f, takes on are called the range of f.

For something to be a function, each domain value has exactly one range value paired with it. It is often said, "every x has only one y." One way to see if something is a function is with the vertical line test. You will need the graph of the expression to use this. It says, a graph is a graph of a function if and only if every vertical line intersects the graph in at most one point. The examples below illustrate this idea.

x<sup>2</sup> + y<sup>2</sup> = 4 y = 2x - 1
x2 + y2 = 4
NOT a function
y = 2x - 1
A function

A function can also be thought of as a set of ordered pairs whose first elements are all different. The set of all first elements of the ordered pairs is the domain of the function. The set of all second elements of the ordered pairs is the range of the function. Consider the set, A={(cat, dog), (chicken, turkey), (cat, hamster)}. A would not be a function because the first element, cat, is paired with 2 different second elements. Consider the set B={(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}. This is a function since each first element has only one second element paired with it. The domain of B is the set {1, 2, 3, 4, 5}, and the range of B is the set {2, 4, 6, 8, 10}. C={(2, 3), (2, 4), (4, 6)} would not be a function since 2 has two elements paired with it, 3 and 4.