For the given functions f and g, find the requested function and state its domain.
-$$f ( x ) = \sqrt { 6 - x } ; g ( x ) = \sqrt { x - 2 }$$
Find $f \cdot g$.
A) $( f \cdot g ) ( x ) = \sqrt { ( 6 - x ) ( x - 2 ) } ; \{ x \mid x \neq 2 , x \neq 6 \}$
B) $( f \cdot g ) ( x ) = \sqrt { ( 6 - x ) ( x - 2 ) } ; \{ x \mid 2 \leq x \leq 6 \}$
C) $( f \cdot g ) ( x ) = \sqrt { ( 6 - x ) ( x - 2 ) } ; \{ x \mid x \geq 0 \}$
D) $( \mathrm { f } \cdot \mathrm { g } ) ( \mathrm { x } ) = \sqrt { - \mathrm { x } ^ { 2 } - 12 } ; \{ \mathrm { x } \mid \mathrm { x } \neq 12 \}$

Question

Quizplus · Accepted Answer

The product of $f(x) = \sqrt{6 - x}$ and $g(x) = \sqrt{x - 2}$ is $\sqrt{(6 - x)(x - 2)}$. The domain is where both radicands are non-negative: $6 - x \geq 0$ and $x - 2 \geq 0$, which simplifies to $x \leq 6$ and $x \geq 2$, respectively. Thus, the domain is $2 \leq x \leq 6$.

For the Given Functions F and G, Find the Requested f(x)=6−x;g(x)=x−2f ( x ) = \sqrt { 6 - x } ; g ( x ) = \sqrt { x - 2 }f(x)=6−x​;g(x)=x−2​

For the Given Functions F and G, Find the Requested $f ( x ) = \sqrt { 6 - x } ; g ( x ) = \sqrt { x - 2 }$