Question 840297
Okay, we first need to find the composition of functions. For the first one you take the function of g(x) and plug it into wherever there is an x in f(x).

When we do this (f*g)(x) becomes y=ln(e^4^x)

Remember, the domain is all of the x-values on a graph. Here is the graph:


{{{ graph( 300, 200, -2, 2, 0, 15, ln(e^4^x)) }}}

As you can tell, the graph will go on in the x and y directions, so the domain is all reall numbers (-infinity,infinity)


Now we will compute (g*f)(x) Using the same method as above; we get y=e^4^(lnx)

Sadly, this program won't let me graph the function, however, if you do it on a calculator or WolframAlpha you will see that the x values will go from negative to positive infinity, again, making the domain all real numbers 
(-infinity,infinity)

Now for (f*h)(x) we will do the same thing as before; we get y=ln(x^3) 

Let's graph it: {{{ graph( 300, 200, -10, 10, -8, 7, ln(x^3)) }}}

Now we see something different, there are no negative x values. This means that the domain is all positive real numbers or (0,infinity).

I hope this helped you!