Question

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

Answer 1

Step 1. The two functions we have are:

$\begin{gathered} f(x)=2x^2 \\ g(x)=\sqrt[]{x-2} \end{gathered}$

And we are asked to find the composite function f(g(x)) and the domain.

Step 2. The function that we need to find is:

$f(g(x))$

To find this, we substitute g(x) into the x value of f(x):

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

Step 3. Simplifying:

The square root and the power of two cancel each other

$f(g(x))=2(x-2)^{}-1$

Distributing the multiplication by 2:

$f(g(x))=2x-4-1$

Combining the like terms:

$f(g(x))=\boxed{2x-5}$

Step 4. Find the domain. The domain is the set of possible values that the x variable can take.

Remember the two original functions:

$\begin{gathered} f(x)=2x^2 \\ g(x)=\sqrt[]{x-2} \end{gathered}$

for f(x) x can take any value. But for g(x) the square root cannot be a negative number, therefore, x-2 has to be equal to or greater than 0:

$\begin{gathered} \text{Domain:} \\ x-2\ge0 \end{gathered}$

Solving for x:

$\begin{gathered} \text{Domain:} \\ x\ge2 \end{gathered}$

This domain also applies to the composite function f(g(x)), and it can be written as follows:

$D\colon\mleft\lbrace x|x\ge2\mright\rbrace$

Answer: Option four

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