 So let's find the domain of the function g of x equals square root of x squared plus x, and let's write our solution using interval notation and graph it on the real number line. So remember the domain consists of all allowable x values. Since we're taking the square root of something, then our radicand must be non-negative, so we require that x squared plus x be greater than or equal to zero. And to solve the inequality, we'll solve the corresponding equality x squared plus x equals zero. Since this is a quadratic equation, we could solve it using the quadratic formula, but this is one of those very, very, very, very, very, very, very, very, very rare cases where it's easy to factor. So we'll factor, and since we have a product equal to zero, one of the factors must be equal to zero. So either x is equal to zero or x plus one is equal to zero. So we'll solve the left-hand side for x, done. Solving the right-hand side for x gives us, and these were the solutions to the equality, so they correspond to the critical points of our inequality, x equals zero and x equals negative one. So we'll graph the critical values, and since the critical values solve the equality x squared plus x equals zero, and we can take the square root of zero, the critical values are included, and so we should shade them in. So we see the critical values divide the real number line into three intervals, so we'll choose a test point in each interval. On the right-hand side, we'll let x equals one million, so then x squared plus x will be a very large positive number, which we can definitely take the square root of, and so we should include the right interval. In the center, we'll have to pick a point, and because zero is a critical value, we can't use zero. Because negative one is a critical value, we can't use negative one. We have to pick something in the middle, so we'll take x equals, so how about negative one-half? So equals means replaceable. If x equals negative one-half, then x squared plus x will be negative one-half squared plus negative one-half, and we can calculate the value of that. And remember, we're actually trying to take the square root of x squared plus x, but because it's negative one-half, we can't take the square root of a negative, so we should exclude this middle interval. Finally, this third interval on the left will let x be negative one million. Let's take a look at our radicand. x squared is going to be the square of a large negative number, so it's going to be a large positive number. And while x will be large and negative, it will be smaller than x squared is going to be. So that means x squared plus x will still be large and positive, and we can take its square root, so the left interval should also be included. And once we have the graph of the domain, we can write our domain in interval notation. Here on the left, we go from the way, way, way, way, way, way left, that's minus infinity, all the way up to negative one, which we include. We don't include the middle, and on the right, we start at zero, include it, and go to the way, way, way, way, way, way right off to infinity. And since our domain consists of both intervals, we'll combine these two with a union symbol.