We say a function is defined for when is a real number. [We will talk about (6)–(11) in more detail in Chapter 3. If you have no prior knowledge about them, you may skip them.]
1. where is any positive integer is defined for every value of .
2. is defined for every value of except , since cannot be calculated.
3. is defined for [because for , is imaginary].
4. where is an even integer is defined for [because for is imaginary].
5. where is an odd integer is defined for every value of .
6. where is defined only for . For negative values of , this function does not exist [in fact, it becomes imaginary].
7. where ; that is, the exponent is a variable and the number being a constant. This function is defined for every value of .
8. or is defined for every value of .
9. is defined for every value of except where is any integer.Recall that . Because , and division by zero is not allowed, these values must be excluded here.
10. or is defined only for .Because sines and cosines cannot exceed +1 or become less than –1, it follows that these functions are defined for all values of ranging from –1 to +1 inclusive, but not for other values.
11. or is defined for every value of .
When we talk about “the function ” and the domain of the function is not specified, the domain is then assumed to be the set of all real numbers for which is defined; that is, is real. This set is sometimes called the natural domain of and is denoted by .
The set of all real numbers for which is real is called the natural domain or simply the domain of the function.
If we want to restrict the domain of a function, we must specify so. For example, we must write “”, to restrict the domain of to positive numbers. If we do not add “”, the domain of is understood to be the entire set of real numbers.
Example 1
Determine the natural domains of the following functions:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Solution
(a) For all , is a real number. Thus .
(b)We recall that is real when . Thus is defined when or . That is,
(c) is defined when or . Thus .
(d) Since or must not be negative, and must always have the same signs (see the following sign table). is defined for every value of except those between 0 and 6; that is We can rewrite the domain of as .
(e) Since must not be negative, is defined where and have the same signs (see the following sign table). Therefore, is defined for , or .
(f) For to be real, we must have or . For to be real, we must have or . Thus, is real, if we have both and ; that is, the domain of is .
(g) Because , is defined for all values of except and . When or , the divisor becomes zero and division by zero is not allowed. Thus . The domain of can also be represented by or .
(h) Because the logarithm of a nonpositive number is not real, we must have or . Thus .
Another important set that is associated with every function is the range of a function. The range of a function is the set of all values taken by the function (or equivalently by the dependent variable) when the independent variable varies over the domain. In other words, all possible outputs of a function make up its range. The range of is denoted by .
If , the range of is
Note that the range of is a subset of its co-domain
The difference between domain, codomain and the range of a function is depicted in Figure 3.
Figure 3. Domain, co-domain and range of a function
Example 2
Let and . Find the range of .
Solution
The range of consists of all possible outputs of . Any is a possible output of , because we can find a value of such that (we just need to set or ). Thus, the range of is .
In elementary calculus, we will learn more powerful methods for determining the ranges of functions.