00:01
All right, so this question, we are looking to find f plus g, f minus g, f times g, f divided by g, and then determine the domain for each function in interval notation.
00:10
And here we have f of x and g of x, both functions of x.
00:14
So first, we are looking to add f plus g.
00:17
I'm going to write this as f plus g of x is equal to, and then we're going to add like terms.
00:33
So we have 2x squared plus 4x, plus...
00:45
1 over 2x.
00:54
To fully understand what this graph is, we need to multiply by a common factor, so we can simplify this to one fraction.
01:04
So to get a common denominator, we have 1 over 2x.
01:07
We're going to multiply this portion, this part of the function, by 2x over 2x, and this is equal to 1, so we're not actually changing what the value of the function is.
01:20
So if we have 2x squared times 2x, we're going to get 4x to the third over this 2x.
01:34
And then we are multiplying a 4x times 2x, and we're going to get 8x squared over 2x.
01:46
And we add our final 1 over 2x.
01:54
So we can simplify this into 1 fraction, and because we have a common denominator, we have 4x to the 3rd plus 8x squared plus 1.
02:14
Over 2x.
02:18
So the only place we're going to have an unreal answer is where the denominator is equal to 0.
02:22
So we set our denominator equal to 0.
02:25
We have 2x equal to 0.
02:29
Really that 2x cannot equal 0.
02:32
And we know that x cannot equal 0 if we divide each side by 2.
02:37
So we're going to write our function in interval notation and that's going to look like negative infinity to where our function is not real, which is at 0, and, that's the sign for and, 0 to positive infinity.
02:56
And this shows that it is not exist at 0, but continues on everywhere else.
03:03
All right, next, we're looking at f minus g.
03:08
F minus g of x is equal to 2x squared plus 4x.
03:21
We'll even that one positive, and we're subtracting this 1 over 2x.
03:38
Once again, we're going to need to simplify this.
03:40
We don't like leaving just one fraction.
03:43
So we're doing the same thing as we did last time, multiplying this whole piece by 2x over 2x.
03:50
That is equal to 1.
03:51
Does not change the value of the function.
03:54
So we just did this so we know what i can copy this down.
03:58
2x squared times 2x over 2x is getting us a 4x to the 3rd over 2x.
04:09
This positive 4x times 2x over 2x is going to give us an 8x squared over 2x.
04:13
But instead we're getting a minus 1 over 2x.
04:26
So we'll find this to 1 fraction.
04:29
We get that f minus g of x is equal to 4x cubed plus 8x squared minus 1 over 2x is our final answer.
04:51
But our domain is also going to look identical because we sit there.
04:54
Our denominator is the same as before.
04:56
We know that x cannot equal 0...