Question

I just need to know if the answer is B or D

Answer 1

Firslty, we need to find the function, whcih we can do by substituting one into the other:

`f(g(x))=2(g(x))^2-1=2(\sqrt[]{x-2})^2-1=2(x-2)-1=2x-4-1=2x-5`

Now, for the domain we have more complicated approach.

We first have to find the domain for both f(x) and g(x).

The domain of f(x) is all real number, because it is a quadratic polynomial.

The domain of g(x) is all the real numbers except the x values that makes the square root interior negative, so all real numbers so that:

`\begin{gathered} x-2\ge0 \\ x\ge2 \end{gathered}`

Now, the domain of the composite function f(g(x)) is the set of all the values of x in the domain of g(x) such that the corresponding g(x) is in the domain of f(x).

Since the domain of f(x) is all real numbers and g(x) is a real number function, all the values of g(x) will be in the domain of f(x), so we just need to check the values of x that are in the domain of g(x).

The domain of g(x), as we saw, is:

`D=\mleft\lbrace x\mright|x\ge2\}`

And any value of x will give a g(x) that will be in the domain of f(x) so, the domain of f(g(x)) is:

`D=\mleft\lbrace x\mright|x\ge2\}`

So, the correct alternative is:

`\begin{gathered} f(g(x))=2x-5 \\ D\colon\mleft\lbrace x\mright|x\ge2\} \end{gathered}`