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1.1 Four Ways to Represent a Function
Functions arise whenever one quantity depends on another. Consider the following four situations.
A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A=πr^2. With each positive number r there is associated one value of A, and we say that A is a function of r.

Table 1 World Population
YearPopulation
(millions)
19001650
19101750
19201860
19302070
19402300
19502560
19603040
19703710
19804450
19905280
20006080
20106870

B. The human population of the world P depends on the time t. Table 1 gives estimates of the world population P at time t, for certain years. For instance,
P≈2,560,000,000 "when" t=1950
C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known.
D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a., how could you tell?

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FIGURE 1 Vertical ground acceleration during the Northridge earthquake
Each of these examples describes a rule whereby, given a number (r in Example A), another number (A) is assigned. In each case we say that the second number is a function of the first number. If f represents the rule that connects A to r in Example A, then we express this in function notation as A=f(r)., how could you tell?

A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E., how could you tell?

We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f(x) Is the value of f at x and is read “f of x.” The range of f is the set of all possible values of f(x) as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called dependent variable. in Example A, for instance, r is the independent variable and A is the dependent variable., how could you tell?

It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f(x) according to the rule of the function. So we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, if you input a number and press the squaring key, the calculator displays the output, the square of the input. FIGURE 2 Machine diagram for a function f, how could you tell?

Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of D to an element of E. The arrow indicates that f(x) is associated with x,f(a) is associated with a, and so on. FIGURE 3
Arrow diagram for f
, how could you tell?

Perhaps the most useful method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs
{(x,f(x)) | x∈D}
(Notice that these are input-output pairs.) In other words, the graph of f consists of all points (x,y) in the coordinate plane such that y=f(x) and x is in the domain of f. The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the y-coordinate of any point (x,y) on the graph is y=f(x), we can read the value of f(x) from the graph as being the heigh tof the graph above the point x. (See Figure 4.) The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5.
, how could you tell?