SOLUTION: How do you find the domain of (g o f)(x) for f(x)=x/x-2 and g(x)= 2x-4/x?

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Question 280605: How do you find the domain of (g o f)(x) for f(x)=x/x-2 and g(x)= 2x-4/x?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(Please put parentheses around numerators and denominators in the future.)

Assuming your functions are:
f%28x%29+=+x%2F%28x-2%29 and g%28x%29+=+%282x-4%29%2Fx then
(g o f)(x) means g(f(x)). So

The domain will be all Real numbers except those that make any of these denominators zero.

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.

The "big" denominator is x%2F%28x-2%29. 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%2F%28x-2%29 equal to zero. (If this is not clear, then set x%2F%28x-2%29+=+0 and solve.) SO we must exclude x=0 from the domain, too.

So the domain of g(f(x)) is all Real numbers except 0 and 2.