 In this video we're going to find the domain of the function f of x equals the square root of x squared minus 5x minus 14 Now because our function f it's a square root of actually there's two functions in play right here So we have our outer function, which is the square root of some function some that number Let's call it you we're composing this with the quadratic polynomial x squared minus 5x minus 14 We've put a quadratic function inside of a square root now a quadratic quadratic has no restriction on its domain you can take any real numbers, but the square root does have some issues, right? We can only take square roots of non-negative numbers. So in order to Compute the domain of this function. We essentially just have to look at the radicand of our square root And that thing needs to be greater than equal to zero So that's the inequality. We now have to solve right now x squared minus 5x minus 14 is Greater than or equal to zero zero is okay because the square root of zero is zero here And so in trying to solve that we solve this quadratic inequality Like we would any other quadratic inequality we could factor we could complete the square We could use the quadratic formula this one factors easy enough factors of negative 14 that out to negative 5 You take x minus 7 and x plus 2 This is great equal to zero and so therefore our markers are going to be 7 and negative 2 I Can place these markers on the x-axis? so I have like a negative 2 right here and I have a positive 7 right here if I want to be greater than or equal to zero that means I'm looking for those places which are above or on the x-axis and What part of a parabola is going to be above the x-axis because the leading coefficient here is a positive one This means the graph would be concave upward. So if you just kind of think of your usual parabola It's gonna be concave upward. It's gonna go up something like this So you have your negative 2 over here your 7 right here the things that are above the x-axis Is going to be the wings of our bird the left wing and the right wing so we're looking at these portions right here And so that then tells us that the domain of our function would be negative infinity to negative 2 2 is included in that Because if I plug in x equals 2, I'm gonna get the square root of 0 which is 0 And then there's a big gap where there's nothing the function is not defined for this region because we're taking the square root of negative numbers Which are not real numbers and then you're gonna union once you get past that that desert in the domain You'll then pick up where you left off 7 to infinity where 7 will be included inside of the domain The markers are included because the square root of 0 is Included now. I do want to mention that if you kind of if you change this problem up Let's say that g of x equals 1 over the square root of x squared minus 5x minus 14 Everything would basically be the same but you have to solve the inequality x squared minus 5x minus 14 is greater than 0 We no longer would allow equal to 0 because if you took if you actually got equal to 0 You take the square of 0, which is 0 That's fine, but then you divide it by 0 so it's undefined So in that situation the domain of g would be negative infinity to negative 2 Union 7 to infinity but this time the the end points negative 2 and 7 would not be included