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