1.4 The Domain and Range of a Function

Consider the function .

Notice that there is no output number when the input number equals 2, since division by zero is meaningless. Thus x can have any value except 2. We say that 2 is not in the domain of the function f.

Here is another example. Let . In this course, we will only consider real-valued functions. This means that no output numbers may contain a multiple of , or the imaginary number i. Thus we cannot allow an input number such as 0 for the function g, since . So zero is not in the domain of g. Furthermore, no value of x for which x < 4 is in the domain of g. [Why is this true?]

The implicit domain principle: It will be assumed that the domain of a function contains every possible input number unless otherwise specified. Only inputs which imply division by zero or which would produce an output containing the square root of −1 are excluded.

For example, the implicit domain of g is the interval [4,). [Why is this true?]

However, it is always possible to restrict the domain of a function to be something less than the implicit domain.

For example, one could define for x > 3. Then the explicit domain is the interval (3,). If it were not for the explicit restriction on the domain of p it would have an implicit domain of (-,).

For each of the following exercises find the implicit domain of the function. Write the domains in interval notation.

Exercise 1.4.1

Exercise 1.4.2

Exercise 1.4.3

We call the set of all inputs of a function its domain, and we call the set of all outputs of a function its range.

It is much easier, in general, to look at the equation of a function and figure out its domain than it is to figure out its range.

For example, take . We can see that its domain is all real numbers except 3. In interval notation that is written . It is not as easy to see what the the range must be. One technique which sometimes works is to replace the in the equation with y and solve the equation for x. When we do this with this example, we find that . Thus we see that the output number y can be anything except 1. Thus, the range of the function is .

Exercise 1.4.4

Find the domain and the range of . Express the answers in interval notation.

Another technique for finding the range is to sketch the graph and see what kind of y values points on the graph may have. For example, if we graph , we see that all the y coordinates of points on the graph are greater than or equal to zero. So the range is .

Exercise 1.4.5

Sketch the graph of . Find its domain and its range.