Problem

Find the domain of the function

Solution

First, we remember that the domain is the set of "allowable" values that we can plug in for x. Why wouldn't we be able to plug in certain values for x? Well, looking at the function f(x), notice that it is a fraction. Since we are not allowed to divide by zero, we must make sure that we exclude the values of x that make the denominator zero.

Since

when

or

That is,

or

When specifying the domain, we must make sure to include the conditions

and

since we want to exclude these values from the domain.

Are there any other restrictions to consider?

Looking at f(x) again, notice that the numerator contains a square root. Since the domain of a square root is nonnegative numbers ("you can't take the square root of a negative number"), we must find the values of x for which x² – 4 is nonnegative.

Solve:

Remember absolute value bars when taking square roots!

or

Putting all these conditions together, we get that the domain of f(x) is the set of all values x such that

or

These conditions guarantee that we will not divide by zero and we will not take the square root of a negative number when calculating f(x).