Step 1 :Given functions are \(f(x)=\frac{x}{x-3}\) and \(g(x)=\frac{5}{x}\)
Step 2 :The composition of two functions, \(f \circ g\), is defined as \(f(g(x))\)
Step 3 :Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))=f\left(\frac{5}{x}\right)=\frac{\frac{5}{x}}{\frac{5}{x}-3}\)
Step 4 :Multiply the numerator and the denominator by \(x\) to simplify the expression: \(f(g(x))=\frac{5}{5-x^2}\)
Step 5 :The domain of this function is all real numbers except for the values that make the denominator equal to zero
Step 6 :Solve \(5-x^2=0\) to find the values to exclude from the domain: \(x^2=5 \Rightarrow x=\pm\sqrt{5}\)
Step 7 :\(\boxed{f(g(x))=\frac{5}{5-x^2}}\)
Step 8 :Therefore, the domain of \(f \circ g\) is \((-\infty,-\sqrt{5}) \cup (-\sqrt{5},\sqrt{5}) \cup (\sqrt{5},\infty)\)