00:01
Alright guys, so we're looking at problem 27 here.
00:04
So the problem which is asking for is to find the domain of a function here function f of p.
00:10
So the function that we get is this here is the function of p is equal to the square root of 2 minus the square root of p so this is a pretty straightforward problem as long as you know what you need to look at.
00:23
So the first thing that we want to do is do a little review of what a domain is so looking at the domain we're going to draw a little graph here so we can get a little an idea of what we're looking at.
00:32
So here's our graph, just any function.
00:35
So we're going to say that this here is some function, a function of x.
00:41
We're going to ignore the fact that this up here is a p.
00:43
We're going to say that this is x to differentiate it because this is not the same function that we have up here is just an example.
00:48
So we're looking at this function.
00:51
When we're looking at the domain, what the domain is, is it's all of the values of x that we can have.
00:56
So we're looking at if we pick a point right here, there is an x, at that point.
01:02
So then this checks out.
01:03
If we pick a point right here, there's an x here.
01:06
That checks out.
01:07
But if we pick a point on the x -axis here, then there's no x there, and we can keep going up forever and ever.
01:14
So that doesn't fit within the domain.
01:15
The domain has to fit within this part of the graph and this part of the graph, so we know what our x values are that work.
01:24
Okay, so that's what we're looking for in this function up here is we're looking for all the values of p that stay within the bounds of what we were given.
01:32
And in this case, the bounds that we were given, specifically, were the square roots.
01:36
So the first thing that we're going to do when we look as this, is this, we're going to look at specifically the variable within the function itself.
01:45
So we want to focus on p.
01:47
I'm going to rewrite the function down here, so we have a little reference to look at.
01:50
So we have the function of p is equal to the square root of two minus the square root of p.
01:58
So what's important here is this p, right? the two, two, we don't quite care about yet.
02:03
The important part is the p.
02:05
So the first step to solving this particular problem is we're going to isolate that square root of p right there.
02:09
We're going to take that out, we're going to look at what that specific value can be.
02:13
So we're going to take this all the way down to here, square root of p.
02:21
So what do we know about square roots? well, we know that anything within the square root cannot be negative.
02:27
We have seen this or else it turns into a kind of a complex number, which is for what further math courses, we don't need to deal with that yet.
02:34
So we're going to say that p has to be greater than zero or equal to zero because the square root of zero is a valid option here because it's just equal to zero.
02:43
So if you look at this we say this can't be negative.
02:51
Therefore we know that zero is the least that p can be.
02:57
But what's the most that p can be? so when we look at this, is there any limit to what a square root can have underneath it? no, there isn't.
03:04
We could conceivably go up as high as we wanted, and there would still be a square root of that number.
03:09
So then we're going to go and we're going to say the greatest that p can be, we're going to put an equal sign there, is infinity.
03:17
Beautiful.
03:19
So just looking at this little bit right here, we know that this is the information that we've gained from isolating the square root of p out of the function.
03:28
So now we're going to go and evaluate the full function after just looking at the full.
03:32
Little piece to see how this plays into the overall function itself.
03:36
So again, i'm going to rewrite the function here.
03:38
The function of p is equal to the square root of two minus the square root of p.
03:45
So the first thing that you want to look at is you want to look at the extremes of what p can be that we found right here.
03:51
So we're going to take our options here, our zero and our infinity, and we're going to plug them in to this equation so that we can see if the equation has more limitations because we started on more of smaller scale we started with our p here that was our smaller scale we figured out the limitations to that part of the problem and now we're going to take that and apply it to the problem or the function as a whole and we're going to see what limitations that may place that are more specific than what we got just with the square root of p so let's plug our numbers in okay so if we plug in we'll start with zero so the function of zero is equal to what the square root of two minus the square root of zero, which is equal to square root of two minus zero, equal to the square root of two.
04:38
Interesting.
04:39
So that fits.
04:41
That works.
04:41
We have a valid option for that function.
04:45
Okay? so zero works as a limiting factor for p.
04:48
We know this.
04:49
This checks out.
04:50
It's not some weird crazy number that we don't know how to deal with.
04:53
It's just the square root of two.
04:54
So we know that zero works.
04:56
Okay, so we're going to write that in yellow here because that's kind of fun.
04:58
Zero.
04:59
Cool.
05:00
That works.
05:01
Okay.
05:01
So now let's go back up and let's look at our other option, infinity.
05:04
Let's plug that in...