1.1 Functions from the Numerical and Algebraic Viewpoints

(This topic is also in Section 1.1 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)

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Let us start by looking at the definition in the textbook (and also in the Chapter 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 (discussed in the next tutorial.)

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.

Press here to link to a page that will allow you to evaluate and graph functions on-line.
Press here to download an Excel grapher. (For the Excel grapher to work, you need to "Enable Macros" when opening the page.)

Examples

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

x	0	1	2	3
f(x)	3.01	-1.03	2.22	0.01

Then, f(0) is the value of the function when x = 0. From the table, we obtain

f(0) = 3.01		Look on the table where x = 0
f(1) = -1.03		Look on the table where x = 1

and so on.

An Algebraically Specified Function:
Suppose that the function f is specified by f(x) = 3x² - 4x + 1. Then

f(2)	= 3(2)² - 4(2) + 1	Substitute 2 for x
	= 12 - 8 + 1 = 5
f(-1)	= 3(-1)² - 4(-1) + 1	Substitute -1 for x
	= 3 + 4 + 1 = 8

Note: Since f(x) is defined for every x, the domain of f is the set of all real numbers.

The following table gives the (approximate) spending on agriculture research in 1976-1996 by the public sector.^*

Year t	6 (1976)	8	10	12	14	16	18	20	22	24	26 (1996)
Expenditure E (Billions)	$2.5	3.0	3.0	3.1	3.0	3.1	3.2	3.3	3.4	3.3	3.1

* Data are approximate. Sources: Economic Research Service, United States Department of Agriculture, American Association for the Advencement of Science, "National Plant Breeding Study 1996" by Dr. Kenneth J. Frey, Iowa State University/New York Times, May 15, 2001, p. F1.

Take E(t) = Expenditure in year t.

E(12) - E(10) =

Q The above answer tells us that spending on agriculture research.

Here once again is the table of values for the function E.

Year t	6 (1976)	8	10	12	14	16	18	20	22	24	26 (1996)
Expenditure E (Billions)	$2.5	3.0	3.0	3.1	3.0	3.1	3.2	3.3	3.4	3.3	3.1

Q Which of the following models best fits the given data. (Use technology like Excel, a graphing calculator, or Function Evaluator & Grapher to compare their values.)

	E(t) = 2.3 + .03t			E(t) = 2.3 - .03t
	E(t) = -0.002t² + 0.1t + 2			E(t) = 0.002t² - 0.1t + 2
	E(t) = 0.02t² - 3.4t + 2

Suppose that the function f is specified algebraically by the formula

f(x) =

x² + 1

with domain [-1, 10).

The domain restriction means that we require -1 x < 10 in order for f(x) to be defined (the square bracket indicates that -1 is included in the domain, and the round bracket after the 10 indicates that 10 is not included).

Now answer the following questions. Fractions or valid technology notation are permitted.) Type "undefined" if the 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, from Chapter 1 in the book.

The percentage p(t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function.^† (t = 0 represents 1997).

p(t)

10t + 15	if	0 t < 1
15t + 10	if	1 t 4

^†The model is based on data through 2000. Source: J.D. Power Associates/The New York Times, January 25, 2000, p. C1

This notation tells us that we use

the first formula, 10t + 15, if 0 t < 1, or, equivalently, t is in [0, 1)
the second formula, 15t + 10, if 1 t 4, or, equivalently, t is in [1, 4].

Thus, for instance,

p(0.5) = 10(0.5) + 15 = 20		We used the first formula since 0 0.5 < 1 Equivalently, 0.5 is in [0, 1)
p(2) = 15(2) + 10 = 40		We used the second formula since 1 2 4 Equivalently, 2 is in [1, 4]
p(4.1) is undefined		p(t) is only defined if 0 t 4 Equivalently, 4.1 is not in the domain [0, 4]

Here is the formula again.

p(t)

10t + 15	if	0 t < 1
15t + 10	if	1 t 4

Q Here are some for you to try. Type undefined if the given value of t is not in the domain of p.

Now try some of the exercises in Section 1.1 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.

p(0) =
p(5) =
p(1) =
p(4) =