If fx sqrt 8x 8 and gx 6x2 1 solve for the following hx and...

if f(x) = sqrt (8x-8) and g(x) = 6x^2 -1, solve for the following h(x) and their domain with interval notation 1) h(x) = (f o g)(x) and it's domain 2) h(x) = (g o g)(x) and it's domain 3) h(x) = (f o f)(x) and it's domain 4) h(x) = (g o f)(x) and it's domain

Transcript

00:01 Square root of 3x minus 7, and then g of x is 6x squared minus 1.

00:09 We want to find the domain and range, or sorry, just the domain of our h function.

00:15 So for the first one, we're going to have f composite g.

00:21 So that means we're sticking a g function inside of the f function.

00:24 So we're going to have the square root of 3 times 6x squared minus 1 minus 7.

00:31 So that's the square root of 18 x squared that's minus 3 then minus 7 so that's negative 10 so if you have a square root we know that our domain is that our inside piece has to be greater than or equal to 0 so 18 x squared minus 10 greater than or equal to 0 so solving for x squared this is going to be 10 over 18 we can simplify that to be 5 over 9 now taking the square root of both sides then we know that x is going to be between negative square root of 5 over 9 and positive square root of 5 over 9 for b we're looking for the composite g of f so we're just swapping the order so in this case now we're sticking our f function into g so we're going to have three or six times the square root of 3x minus seven squared then minus one where the square root and the square are going to cancel so this is just six times 3x minus seven minus one so here we have no restrictions because this is going to be a basic polynomial so our domain would be all real numbers or you could say negative infinity to positive infinity for c we're doing f composite f so we're just taking our f function into itself so we have the square root of 3 times the square root of 3x minus 7 and then minus 7 so we kind of have a twofold scenario here where we have the inside function which is 3x minus 7 we have that that has to be greater than or equal to 0.

02:35 So x has to be greater than or equal to 7 over 3.

02:39 And then we have the other scenario where we have this inside piece that needs to be greater than or equal to 0.

02:48 So this is just the square root 3x minus 7 greater than or equal to 7 over 3...

Question

if f(x) = sqrt (8x-8) and g(x) = 6x^2 -1, solve for the following h(x) and their domain with interval notation 1) h(x) = (f o g)(x) and it's domain 2) h(x) = (g o g)(x) and it's domain 3) h(x) = (f o f)(x) and it's domain 4) h(x) = (g o f)(x) and it's domain

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