Explain using the theorems why the function is continuous at...

Explain, using the theorems, why the function is continuous at every number in its domain. $Q(x) = \frac{\sqrt{x-6}}{x^3-6}$ $Q(x)$ is a polynomial, so it is continuous at every number in its domain. $Q(x)$ is a rational function, so it is continuous at every number in its domain. $Q(x)$ is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain. $Q(x)$ is not continuous at every number in its domain. none of these State the domain. (Enter your answer using interval notation.) $(6, \infty)$

Transcript

00:01 Okay, we're given this function and they want us to explain why it's continuous everywhere in its domain and then find its domain.

00:09 So first we better make sure we know what a domain is.

00:13 So the domain of a function is the set of all numbers, x, that make a real number come out of the function when you put x in it.

00:26 So numbers that don't cause a problem in the function.

00:31 Okay, so instead of this one, let's look at a couple of other functions first.

00:37 So if i had this, f of x equals the square root of x, you know that you can't plug minus 4 in there because the square root of minus 4 is 2i, and 2i is not real.

00:50 So for this function, minus 4 would not be in the domain.

00:56 In fact, no negative numbers would be in the domain.

00:59 The domain of this is all x greater than or equal to zero.

01:05 Okay, so if there's a square root, you have to find out what makes the stuff underneath greater than or equal to zero.

01:11 I don't know why i start scribbling that out.

01:13 Okay, i is bad when you're trying to find the domain.

01:18 Okay, the other bad thing is, for example, this.

01:21 Let's say we had 20 over x minus 1.

01:26 All right, well, if you tried to plug one in there, you would get 20 over 0.

01:32 Okay, also bad, because that's undefined.

01:37 Okay, so you can't let a function have zero in the denominator.

01:43 Okay, but if you add f of x equals the cube root of x, you could let x be anything.