 In this video, we will provide the solution to question number 22 for the practice final exam for math 1050. We're given a rational function f of x equals x squared plus 3x plus 2 on the top at x squared plus 4x plus 4 in the denominator, and we need to find the domain of f, write an interval notation. Now, to find the domain of a rational function, it really just comes down to what makes the denominator go to zero, all right? So we have to factor the equation x squared plus 4x plus 4, I should say factor the polynomial. We need a set equals zero, that's why we're factoring, right? This one, x squared plus 4x plus 4, this is actually a perfect square trinomial of factors as x plus 2 quantity squared. Take the square root, of course, you get x plus 2 equals zero, so x equals negative 2, that's the forbidden value. And so there's only one value that makes the denominator go to zero, that's negative 2. Interval notation, the domain would be negative infinity to negative 2, union 2 to, negative 2 to infinity, excuse me. And so this is, in fact, the domain of f, we're gonna put a box over here, and that's all they have to do, just look at the denominator. Now there is a trap in this one, because we factor the denominator, what if we factor the numerator? So x squared plus 3x plus 2, that factors as x plus 1 and x plus 2, like so, and you can forward that out again. And the denominator, right, looks like x plus 2 times x plus 2. Now some of us, if you factor the numerator, it might be tempted to try to simplify this thing. So you get x plus 1 over x plus 2, now fortunately in this one, that doesn't change the domain, I mean since the denominator still goes to zero to negative 2, you wouldn't be too confused about it. But if things were a little bit different, again this is an important thing to mention, what if the denominator turned out to be like x minus 2 in this situation? So then we would have said the domain, right, is basically everything except for 2 and negative 2, right? Because that's what makes the denominator go to zero, but if you cancel this out, you might think negative 2 is no longer a problem and only positive 2. The domain of a rational function is always determined by the original denominator. It doesn't matter if it can be simplified. The original denominator is where the restrictions are provided. So if you're finding the domain of a rational function, ignore the numerator and only look at the denominator because even if something cancels the numerator, it don't matter. You only look at the denominator here for the domain.