Drawing Graphs of Functions

A second way to visualize a function is by drawing its graph. The graph of a real function f of one variable is the set of all points P(x, y) in the plane such that y = f(x). To draw the graph, we plot the value of x on the horizontal, or x-axis and the value of f(x) on the vertical, or y-axis. How can we tell whether a set of points in the plane is the graph of some function? By reading the definition of a function again, we have an answer.

A set of points in the plane is the graph of some function f if and only if for each vertical line one of the following happens:

(1)    Exactly one point on the line belongs to the set.

(2)    No point on the line belongs to the set.

A vertical line crossing the x-axis at a point a will meet the set in exactly one point (a, b) if f(a) is defined and f(a) = b, and the line will not meet the set at all if f(a) is undefined. Try this rule out on the sets of points shown in Figure 1.2.2.

Figure 1.2.2

Here are two examples of real functions of one variable. Each function will be described in two ways: the black box approach, where a rule is given for finding the value of the function at each real number, and the graph method, where an equation is given for the graph of the function.

Example 1: The Square Function

Example 2: The Reciprocal Function

In Examples 1 and 2 we have used the variables x and y in order to describe a function. A variable is a letter which stands for an arbitrary real number; that is, it "varies" over the real line. In the equation y = x², the value of y depends on the value of x; for this reason we say that x is the independent variable and y the dependent variable of the equation.

In describing a function, we do not always use x and y; sometimes other variables are more convenient, especially in problems involving several functions. The variable t is often used to denote time.

It is important to distinguish between the symbol f and the expression f(x). f by itself stands for a function. f(x) is called a term and stands for the value of the function at x. The need for this distinction is illustrated in the next example.

Finding Values of Terms by Substitution.