Lesson 1.1: Relations, Functions, and Graphs (Review)

In calculus, we study functions, transformations in which each input or combination of inputs is associated with exactly one output. Functions take one or more inputs, or arguments: a function f that takes one argument might be written as , and a function g that takes three arguments might be written as . Because it is the more broad function, let’s consider the function g, in which x, y, and z represent the independent variables, and outputs one value. Suppose that a room is divided into a Cartesian coordinate system in three dimensions. Then could represent the temperature at a point in that room. In the realm of single-variable calculus, to which this section of the site is devoted, we will deal exclusively with functions having a single input.

Relations are a more generalized case of functions where each input may result in multiple outputs. The difference between relations and functions is easier to understand when we discuss graphical representations of functions.

Frequently, we speak of the graph of a function. Most familiar are graphs of the form , where the horizontal axis represents the input and the vertical axis represents the output. In Chapter 11, we also discuss parametric graphs, where each coordinate is determined by a separate function, and polar graphs of the form .

Many common shapes can be represented by graphs, including lines, circles, and other conic sections. Graphs of functions are identifiable in that they pass the “vertical line test”: any vertical line drawn on the graph will pass through no more than one point on the function’s graph. Graphs of relations, such as ellipses, may fail this test.

Terminology

• A function is a transformation in which each unique input or combination thereof gives exactly one output value.
• The domain of a function is the set of all valid inputs to the function.
• The range of a function is the set of all possible outputs of the function.
• The independent variables of a function are the variables representing the input to the function.
• The dependent variable of a function represents the output of the function.