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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:
Note on Domains
Press here to link to a page that will allow you to evaluate and graph functions on-line.
A Numerically Specified Function:
Then, f(0) is the value of the function when x = 0. From the table, we obtain
and so on.
An Algebraically Specified Function:
Take E(t) = Expenditure in year t.
Suppose that the function f is specified algebraically by the formula
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.
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).
This notation tells us that we use
|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.
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.