 One of the important questions we can ask about functions is to find their domain and again remember that what a function does is it's going to associate a unique output value with a specific input value. So given a particular value as our input our function gives us at most one output value. Now the output value does have to be unique but it's possible that there may actually be no output value at all for any particular input value. In this case there's no output value because our input value may be forbidden. And so one of the questions that we do want to consider is what can we substitute into our function? And so the set of allowable input values the things we can substitute into our function and get some output value these are known as the domain of the function. Well for a variety of reasons in general the easiest way of figuring out what the domain is is to begin by identifying the forbidden values. And once you have those forbidden values then your domain can be everything else. For example let's take the domain of this function here. A little analysis goes a long way. This is a rational function. This is a quotient of two polynomials. Any time you have a quotient the concern that you have is as the denominator ever going to be equal to zero. And so I want to make sure that the denominator is never to be zero. And so what I will do to find the domain I know that's a forbidden value. And so I want to find what makes this equal to zero and then I'll say don't choose these things. Anything else is fine but the values that make the denominator equal to zero. So this is a quadratic equation. So quadratic equation equal to zero. I can apply the quadratic formula. And so I'll substitute those values in A equals one, B equals negative three, C equals positive six. I'll drop those into my quadratic formula. And I'll solve and let the dust settle. And well it turns out my solutions in this case are complex numbers. And we're only concerned with functions of real variables. So since the only thing that will make our quadratic equal to zero is a complex number, no real number will make the denominator zero, so no real number gives us a problem. So none of the real numbers have to be excluded. And so any real number is part of the domain. And so I can say that my domain includes all real numbers. How about a different function? G of x equals square root x squared minus 4x minus 12. And again this is a square root function. And again we're only concerned with real functions of real variables. So here I need to make sure that my radicand, the thing I'm taking the square root of, has to be non-negative. So I want to find the forbidden values. The radicand can't be negative. So we'll see what makes it negative and we'll forbid those. So here's radicand negative. Things are less than zero. And I can solve this. This is an inequality. This is a quadratic. So I don't have any forbidden values for the quadratic itself. And so the critical points are going to be the solutions to the corresponding equation. So I'll apply the quadratic formula. A equals 1, B equals negative 4, C equals negative 12. Drop this into the quadratic formula. And after all the dust settles, So I'll do a little bit of arithmetic. I'll do a little bit of arithmetic. I'll do a little bit of arithmetic. That's 4 plus 8 over 2, otherwise known as 6. And the other solution, 4 minus 8 over 2, otherwise known as negative 2. So my two solutions, my critical points, are going to be x equals 6. And also x equals negative 2. So I'll graph the critical points and I'll choose some suitable test points. And I'll find my solution set. What's going to make this inequality true is this middle section here between negative 2 and 6. And again, I don't want this to happen. If this happens, I'm trying to take the square root of a negative number. So this set here is actually the forbidden set and we should exclude them. Don't take that bit and we want everything else. Now, one thing to note is that we definitely want the stuff out here and out here. Also, negative 2 and 6 were not part of the solution set to this. So when we got rid of the solution set, when we got rid of everything that's going to make our radicand negative, we did not get rid of negative 2. We did not get rid of 6. So those should be part of our domain. So we'll go ahead and shade that in and answer the question in the dialect that it was given in. This is an algebraic formula. This is a geometric picture. We do want to give our function domain in an algebraic form. And so let's see that x is less than or equal to negative 2 or x is greater than or equal to 6. And so there's the domain of my function.