so this is one of those problems I absolutely did not solve after numerous attempts. The relations are already troubling me.. yet they also want me to find the domain and range.. I'm lost =[
a)
For $\displaystyle f(x) \cdot g(x)$, you simply multiply.
$\displaystyle (f \cdot g)(x) = { \left(\dfrac {6 \sqrt{x}+x^{2} }{\sqrt[5]{x-2}}\right)\left(\dfrac{3}{5\sqrt[3]{x^{2}}}\right)} = {\dfrac{3 \left( 6\sqrt{x}+x^{2}\right)}{\left( 5\sqrt[3]{x^{2}}\right) \left( \sqrt[5]{x-2}\right)}} = {\dfrac{3x^{\frac{1}{2}} \left(x^{\frac{3}{2}} + 6\right)}{5x^{\frac{2}{3}}\left(x-2\right)^{\frac{1}{5}}}} =$
$\displaystyle {\dfrac{3\left(\sqrt[3]{x^{2}}+6\right)}{5 \sqrt[6]{x}\sqrt[5]{x-2}}}$.
To find the domain, notice that if the denominator equals 0, it makes the function undefined. So $\displaystyle 5\sqrt[6]{x}\sqrt[5]{x-2}=0$, $\displaystyle x = 0$ or $\displaystyle x = 2$. So thus far, the domain is $\displaystyle x \neq 0$ and $\displaystyle x \neq 2$. But also notice the square root in the denominator, so x cannot be negative. So the domain is $\displaystyle x > 0$ and $\displaystyle x \neq 2$.
The range is a bit trickier. I'm not really sure how to get it either.
Remember the properties of exponents, $\displaystyle x^{\frac{a}{b}} = \sqrt[b]{x^{a}}$, and the distributive property.
From $\displaystyle 3\left(6\sqrt{x}+x^{2}\right)$.
Convert the radical to an exponent,
$\displaystyle 3\left(6x^{\frac{1}{2}}+x^{2}\right)$.
Factor out the $\displaystyle x^{\frac{1}{2}}$,
$\displaystyle 3x^{\frac{1}{2}}\left(x^{\frac{3}{2}} + 6\right)$
C and D are compositions of functions. You might want to check here for reference, Composition of Functions
For C)
$\displaystyle (f \circ g)(x)={f(g(x))} = {\dfrac{6 \sqrt{\dfrac{3}{5\sqrt[3]{x^{2}}}} + \left( \dfrac{3}{5 \sqrt[3]{x^{2}}} \right)^{2}} {\sqrt[5]{5\sqrt[3]{x^{2}}-2}}}$
This gets really really messy; I didn't even bother trying to work this out by hand; but according to my TI-89 calculator, the composition should look something like this:
$\displaystyle \dfrac{3 \sqrt[5]{5} \left( 3 \sqrt[3]{|x|} + 10 \sqrt{15} \sqrt[3]{x^{4}} \right)} {25 \sqrt[5]{x^{6}} \sqrt[5]{10\sqrt[3]{x^{2}}-3}\sqrt[3]{|x|}}$
This is obviously very messy to work with. I would suggest using a graphing calculator to approximate the domain and range.
Good Luck.
But the concept for C and D is really simple. It's just those functions are really messy to begin with.
Say $\displaystyle f(x) = x+2$ and $\displaystyle g(x)= x^{2}$
So
$\displaystyle (f \circ g)(x) = f(g(x)) = f(x^{2}) = x^2 + 2$.
The domain and range of this is trivial.