Answer:
domain: all real numbers except x=3 and x=5.
Step-by-step explanation:
You want the domain of a composition of rational functions f(x)=1/(x-2) and g(x)=4/(x-3).
Domain
The domain of a function is the set of values of the independent variable for which the function is defined.
To compute f(g(x)), you must first compute g(x). When x=3, the denominator of the function 4/(x-3) is zero, so the value is undefined. The domain must exclude x=3.
The composition is found by substituting g(x) for x in the definition of f(x).
[tex]f(g(x)) = f(\dfrac{4}{x-3}) =\dfrac{1}{\dfrac{4}{x-3}-2}=\dfrac{x-3}{4-2(x-3)}\\\\f(g(x))=\dfrac{x-3}{4-2x+6}=\dfrac{x-3}{2(5-x)}[/tex]
We have written the composition in this form so you can see there is a denominator factor that will be zero when x=5. The domain of f(g(x)) must also exclude x=5.
The domain of f(g(x)) is all real numbers except x=3 and x=5.
