Question

two functions f and g are composed such that $f(g(x)) = 4\sqrt{x - 3} + 2$. which of the following could be the expressions for $f(x)$ and $g(x)$?
a $f(x) = \sqrt{x}$ and $g(x) = 4x - 3$
b $f(x) = 4x - 3$ and $g(x) = \sqrt{x}$
c $f(x) = \sqrt{x - 3}$ and $g(x) = 4x + 2$
d $f(x) = 4x + 2$ and $g(x) = \sqrt{x - 3}$

Explanation:

Step1: Recall function composition

Function composition \( f(g(x)) \) means we substitute \( g(x) \) into \( f(x) \). So we need to check each option by substituting \( g(x) \) into \( f(x) \) and see if we get \( 4\sqrt{x - 3}+2 \).

Step2: Check Option A

Given \( f(x)=\sqrt{x} \) and \( g(x)=4x - 3 \). Then \( f(g(x))=f(4x - 3)=\sqrt{4x - 3} \). This is not equal to \( 4\sqrt{x - 3}+2 \), so A is incorrect.

Step3: Check Option B

Given \( f(x)=4x - 3 \) and \( g(x)=\sqrt{x} \). Then \( f(g(x))=f(\sqrt{x}) = 4\sqrt{x}-3 \). This is not equal to \( 4\sqrt{x - 3}+2 \), so B is incorrect.

Step4: Check Option C

Given \( f(x)=\sqrt{x - 3} \) and \( g(x)=4x + 2 \). Then \( f(g(x))=f(4x + 2)=\sqrt{(4x + 2)-3}=\sqrt{4x - 1} \). This is not equal to \( 4\sqrt{x - 3}+2 \), so C is incorrect.

Step5: Check Option D

Given \( f(x)=4x + 2 \) and \( g(x)=\sqrt{x - 3} \). Then \( f(g(x))=f(\sqrt{x - 3})=4(\sqrt{x - 3})+2=4\sqrt{x - 3}+2 \), which matches the given composition. So D is correct.

Answer:

D. \( f(x) = 4x + 2 \) and \( g(x) = \sqrt{x - 3} \)