Question 280605
(Please put parentheses around numerators and denominators in the future.)<br>
Assuming your functions are:
{{{f(x) = x/(x-2)}}} and {{{g(x) = (2x-4)/x}}} then
(g o f)(x) means g(f(x)). So
{{{g(f(x)) = (2(f(x)) - 4)/(f(x)) = (2(x/(x-2)) - 4)/(x/(x-2))}}}
The domain will be all Real numbers except those that make any of these denominators zero.<br>
First let's look at the "little" denominators. They are both x-2. I hope it is clear that x=2 would make x-2 zero. (If not, then set x-2 = 0 and solve.) So we must exclude 2 from the domain.<br>
The "big" denominator is {{{x/(x-2)}}}. This is a fraction and if we understand fractions well we know that they are zero only if the numerator is zero. So x=0 would make {{{x/(x-2)}}} equal to zero. (If this is not clear, then set {{{x/(x-2) = 0}}} and solve.) SO we must exclude x=0 from the domain, too.<br>
So the domain of g(f(x)) is all Real numbers except 0 and 2.