Thank you for the opportunity to help you with your question!
(g o f)(x) is a composite function which can also be written as g(f(x)).
g(f(x))=g(2/x) I plugged in 2/x for f(x) in this equation.
Now g(2/x)=5/[(2/x)-1] I plugged in 2/x for x in the g(x) function.
5/[(2/x)-1]=5/[(2/x)-(x/x)] Here I made 1 into x/x so I could make a common denominator to combine the terms in the denominator of right side of the equation. Now I can combine the fractions.
5/[(2-x)/x] Now when you have a fraction in the denominator to simplify you just multiply by the reciprocal of the fraction.
5x/(2-x) is your answer and to find the domain all you need to do is find where the function will not produce a value. This can happen in only two ways. If you have a 0 in a denominator or a negative in a radical you can not get a real value for a function. Since there is no radical we can ignore that possibility. There is however a denominator which can have a 0. Set the denominator equal to 0.
2-x=0 Add x to each side.
x=2 This is when the denominator equals 0 and thus can not be part of the domain of the function. The domain is the set of all x values which will produce a real number answer when plugged into the function.
The domain is all real numbers except 2. Another way to state the domain is x does not equal 2.
Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 26th, 2015
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