GMAT Exam: A Simple Guide To Algebra - Basic Function Concepts

Terminology
A function is a set of ordered pairs (x, y) such that no two different ordered pairs have the same first coordinate.

The domain of a function is the set of all first coordinates of the ordered pairs in the function.

The range of a function is the set of all second coordinates of the ordered pairs in the function.

For example, the set of ordered pairs f = {(2, 5), (3, 7), (4, 1), (5, 5)} is a function.

The domain of f is D = {2, 3, 4, 5} and its range is R = {1, 5, 7}.

The set of ordered pairs s = {(4, 2), (4, 3), (5, 1), (6, 3)} is not a function, because (4, 2) and (4, 3) have the same first coordinate.

It is common to use an equation to define a function.

For example, the equation y = 3x + 7 specifies how to obtain the ordered pairs (x, y) for a function.

As you substitute values for x into y = 3x + 7, you obtain corresponding values for y.

Thus, for instance, (–2, 1), (1, 10), (2, 13), (3, 16), and (4, 19) are ordered pairs in the function.

You can refer to x as the independent variable and to y as the dependent variable.

A common notation for functions is to use the symbol f(x) to denote the value of the function f at a given value for x.

Tip: The notation f(x) does not mean f times x.

In this setting, it is convenient to designate x as the input value and f(x) as the output value.

In terms of ordered pairs, if f = {(2, 5), (3, 7), (4, 1), (5, 5)} then f(2) = 5, f(3) = 7, f(4) = 1, and f(5) = 5.

You can express y = 3x + 7 as f(x) = 3x + 7, where y = f(x).

Tip: Even though a function f is a set of ordered pairs, it has become commonplace to refer to an equation that defines a function as the function; that is, to speak of “the function y = 3x + 7” or “the function f (x) = 3x + 7.”

Evaluating Functions
To evaluate a function, replace the function’s variable with the indicated number or expression.

Here are examples.


Some Common Functions
Here are some common functions you might see on the GMAT.

Tip: The graph of a rational function might have vertical asympotes.

A vertical asymptote is a vertical line that corresponds to a value for the variable that produces zero in the denominator of a simplified rational function.