Problem

Consider the functions

and determine the domains of fog and gof.

Solution

First,

The domain will consist of all values of x such that we aren't taking the square root of a negative number. So x must satisfy two conditions:

(so the inner square root is nonnegative)

and

(so the outer square root is nonnegative)

But if x satisfies the latter condition, it also satisfies the former as well, so we can simply write the domain of fog(x) as all x such that

Next,

As before, x must satisfy two conditions:

(so the inner square root is nonnegative)

and

(so the outer square root is nonnegative)

If x satisfies the latter condition, it also satisfies the former, so the domain of gof(x) is all x such that