Math Help - relations problem

1. relations problem

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 =[

2. a)

For $f(x) \cdot g(x)$, you simply multiply.

$(f \cdot g)(x) = { \left(\dfrac {6 \sqrt{x}+x^{2} }{\sqrt[5]{x-2}}\right)\left(\dfrac{3}{5\sqrt[3]{x^{2}}}\right)} = {\dfrac{3 \left( 6\sqrt{x}+x^{2}\right)}{\left( 5\sqrt[3]{x^{2}}\right) \left( \sqrt[5]{x-2}\right)}} = {\dfrac{3x^{\frac{1}{2}} \left(x^{\frac{3}{2}} + 6\right)}{5x^{\frac{2}{3}}\left(x-2\right)^{\frac{1}{5}}}} =$
${\dfrac{3\left(\sqrt[3]{x^{2}}+6\right)}{5 \sqrt[6]{x}\sqrt[5]{x-2}}}$.

To find the domain, notice that if the denominator equals 0, it makes the function undefined. So $5\sqrt[6]{x}\sqrt[5]{x-2}=0$, $x = 0$ or $x = 2$. So thus far, the domain is $x \neq 0$ and $x \neq 2$. But also notice the square root in the denominator, so x cannot be negative. So the domain is $x > 0$ and $x \neq 2$.

The range is a bit trickier. I'm not really sure how to get it either.

3. After you did the 2nd step, I don't know how you got 3x^1/2(x^3/2 +6). I also can't figure out what you did after that ^.^''

Thanks

4. Remember the properties of exponents, $x^{\frac{a}{b}} = \sqrt[b]{x^{a}}$, and the distributive property.

From $3\left(6\sqrt{x}+x^{2}\right)$.

Convert the radical to an exponent,
$3\left(6x^{\frac{1}{2}}+x^{2}\right)$.

Factor out the $x^{\frac{1}{2}}$,
$3x^{\frac{1}{2}}\left(x^{\frac{3}{2}} + 6\right)$

5. Thank you so much! Yay I have 1/4 of this problem solved haha. Does anyone have any ideas on how to solve the other parts of this problem? (especially c and d)

6. C and D are compositions of functions. You might want to check here for reference, Composition of Functions

For C)

$(f \circ g)(x)={f(g(x))} = {\dfrac{6 \sqrt{\dfrac{3}{5\sqrt[3]{x^{2}}}} + \left( \dfrac{3}{5 \sqrt[3]{x^{2}}} \right)^{2}} {\sqrt[5]{5\sqrt[3]{x^{2}}-2}}}$

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:

$\dfrac{3 \sqrt[5]{5} \left( 3 \sqrt[3]{|x|} + 10 \sqrt{15} \sqrt[3]{x^{4}} \right)} {25 \sqrt[5]{x^{6}} \sqrt[5]{10\sqrt[3]{x^{2}}-3}\sqrt[3]{|x|}}$

This is obviously very messy to work with. I would suggest using a graphing calculator to approximate the domain and range.

Good Luck.

7. Wow.. that does look very messy O_O .. I'll just go as far as i can.. haha

Thanks for the help!

8. But the concept for C and D is really simple. It's just those functions are really messy to begin with.

Say $f(x) = x+2$ and $g(x)= x^{2}$

So
$(f \circ g)(x) = f(g(x)) = f(x^{2}) = x^2 + 2$.

The domain and range of this is trivial.

9. I'm pretty familiar with that concept =] This problem is just the toughest i've seen.