Functions from the Numerical, Algebraic, and Geometric Viewpoints

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Let us start by looking at the definition in the textbook (and also in the topic summary)
Functions and Domains

A real-valued function f of a real variable is a rule that assigns to each real number x in a specified set of numbers, called the domain of f, a single real number f(x).

The variable x is called the independent variable. If y = f(x) we call y the dependent variable.

A function can be specified:

  • numerically: By means of a table
  • algebraically: By means of a formula
  • graphcially: By means of a graph

The graph of a function consists of all points (x, f(x)) in the plane with x in the domain of f.

Note on Domains
The domain of a function is not always specified explicitly; if no domain is specified for the function f, we take the domain to be the largest set of numbers x for which f(x) makes sense. This "largest possible domain" is sometimes called the natural domain.

A Function Specified Numerically
Suppose that the function f is specified by the following table:


By f(0) we mean the value of the function f when x = 0, and by f(1), the value of the function when x = 1, and so on. From the table, we obtain

    f(0) =     Look on the table where x = 0
    f(1) =     Look on the table where x = 1

and so on.

Evaluate the following:
    f(2) =
    f(2) + f(1) =
    f(2+1) =

Here is the graph of f, obtained by plotting the points in the table:

Use the graph to estimate the following (click on the graph and use the arrow buttons below the graph to trace):

Note These estimates are called interpolations as they estimate the value of a function at a point by knowing the values at points on both sides.

A Function Specified Graphically
The following graph shows the total population in state and federal prisons in 1970-1997 as a function of time in years (t = 0 represents 1970).*

* Data are approximate. Sources: Bureau of Justice Statistics, New York State Dept. of Correctional Services/The New York Times, January 9, 2000, p. WK3.

To estimate a value P(a) from the graph, find the y-coordinate of the point where x = a. Here is the graph again, showing estimates of P(20) and P(25):

P(20) = 0.8       0.8 million people in prison in 1990 (t = 20)
P(25) ≈ 1.2       1.2 million people in prison in 1995 (t = 25)

An example for you to try:
The following graph shows daily quesadilla sales S(t) at your fast food store as a function of time t in days:

        Quesadilla Sales
Use the graph to estimate the following (click on the graph and use the arrow buttons below the graph to trace):
The second answer above tells you that sales of quesadillas

An Algebraically Specified Function
Suppose that the function f is specified by Then

    f(2) =   Substitute 2 for x.
    f(−1) =   Substitute −1 for x.
Evaluate the following:
Note: Since f(x) is defined for every x, the domain of f in this example is the set of all real numbers.
Which of the following is the graph of the function f specified above by (Click on the correct graph.)

Suppose that the function f is specified algebraically by the formula

The domain restriction means that we require in order for f(x) to be defined (the bracket indicates that included in the domain, and the bracket after the bracket indicates that included).

Now evaluate the given expressions. Fractions, decimals accurate to three decimal places, or valid technology formulas are permitted. Type if the function is not defined at the given value of x.

Piecewise Defined Functions

Sometimes we need more than a single formula to specify a function algebraically, as in the following example, similar to Example 2 in the textbook:

The number, in millions, of Facebook members from 2004 to 2009 can be approximated by the following function (t = 0 represents the start of 2004):
Sources for data: and

Thus, for instance,
We use the first formula since 0 ≤ 2.5 ≤ 3.
We use the second formula since 3 < 3.5 ≤ 5.
n(5.5) is undefined. The domain of n is [0, 5].

For the following questions, type if the function is not defined at the associated value of t. Here is the function again:

Here is another piecewise defined function; this time with three formulas (notice that its graph now consists of three pieces):

You now have several options:

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Last Updated: November, 2009
Copyright © 2009 Stefan Waner