Question

If the domain of each of the functions f(x) and g(x) is all real numbers, will the domain of (f/g)(x) also be all real numbers? Why or why not?

Answer 1

You can't make any conclusion from the information provided. For instance, suppose that g(x)=2^x. The range is all positive numbers so the domain of (f/g)(x) would also be all real numbers.

But now suppose g(x)=2x the domain is all real numbers and so is the range however (f/g)(x) can't include x=0 because g(0)=0 and you can't divide by zero!

So you really can't say anything about what the domain of (f/g)(x) is except that the range of g(x) can't contain 0.