 So let's try to find the domain of a composite function, let f of x equals 1 over x plus 1, and g of x equals 1 over x. Let's find f of g of x and state its domain. And so we want to find f of g of x. Well, remember this notation really means f of g of x. Well, equals means replaceable, and g of x is 1 over x. So we'll replace g of x with 1 over x. And so what we want to know is what's f of 1 over x. So we know that f of x is 1 over x plus 1. But if we want to say anything about the composite function, what we want to do is we want to drop every occurrence of x and replace it with an empty set of parentheses. And because paper is cheap, we'll copy that down. And the important thing to keep in mind is that whatever goes in one set of parentheses has to go in all the sets of parentheses. So the thing we want here is 1 over x. So we'll put a 1 over x here, as well as every other place we see an empty set of parentheses right here. E equals means replaceable. So f of 1 over x is equal to 1 over 1 over x plus 1. So we'll replace. And that gives us our composite function f of g of x. What about our domain? The thing to recognize here is that our function is a mess of fractions. And so remember that we require that no denominator can be 0. So the thing to recognize here is we actually have two denominators. We have this little tiny denominator here, which cannot be 0. And so we require that x can't be 0. And then this whole thing, 1 over x plus 1, is also a denominator. And so we require that 1 over x plus 1 can't be 0. And remember to solve the inequality, solve the corresponding equality. So our first inequality is x cannot be equal to 0. So we'll solve the corresponding equality, x equals 0. So we'll solve it. And since this is the thing that will make a denominator equal to 0, we require that x cannot be equal to 0. The other inequality, 1 over x plus 1, we want to solve the inequality by solving the corresponding equality. So we'll solve the equation 1 over x plus 1 equals 0. And because this has a fraction, we'll multiply by the denominator of the fraction. So we'll multiply both sides by x, we'll expand, we'll simplify. And that gives us a nice equation, 1 plus x equals 0. Since the left-hand side is a sum, we can get rid of it by subtracting 1 from both sides. And we get our solution, x equal to negative 1. So again, this solves the equality, but we don't want this to happen. So we also require that x cannot be negative 1. And so here's our function. And our domain is x not equal to 0, x not equal to negative 1.