Function and Their Representation, Lecture notes of Mathematics

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FUNCTIONS AND LIMITS

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Functions and

Their Representations

1.

Functions and Their Representations B. The human population of the world P depends on the time t. The table gives estimates of the world population P ( t ) at time t , for certain years. For instance, P (1950)  2,560,000, But for each value of the time t there is a corresponding value of P , and we say that P is a function of t.

Functions and Their Representations Both examples describes a rule whereby, given a number ( r , t ), another number ( A , P ) is assigned. In each case we say that the second number is a function of the first number. 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 .”

It’s helpful to think of a function as a machine (see Figure 2). Machine diagram for a function ƒ Figure 2 Functions and Their Representations

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. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. Functions and Their Representations

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. The most common 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 . The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Functions and Their Representations

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 height of the graph above the point x. (See Figure 4.) Figure 4 Functions and Their Representations

Example 1

The graph of a function f is shown in Figure 6. (a) Find the values of f (1) and f (5). (b) What are the domain and range of f? Figure 6

Representations of Functions

Representations of Functions

The table of values of world population provides a convenient representation of this function.

Representations of Functions

If we plot these values, we get the graph (called a scatter plot ) in Figure 7. Figure 7 Scatter plot of data points for population growth

Representations of Functions

Figure 8 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. Figure 8 Graph of a mathematical model for population growth

Representations of Functions

In other words, it is a function with an explicit formula that approximates the behavior of our given function. The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientific experiment.