Find the composite function (gof)(x) and state its domain

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Let f(x) = 2/xand g(x) =5/x-1. Find the composite function (g o f)(x)and state its domain.

Oct 26th, 2015

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.

5*x/(2-x)=5x/(2-x)

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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