O Functions
Composition of two rational functions
Alexander
For the functions $f(x)=\frac{x}{x-3}$ and $g(x)=\frac{5}{x}$, find the composition $f \circ g$ and simplify your answer as much as possible. Write the domain using interval notation.
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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 \((-\infty,-\sqrt{5}) \cup (-\sqrt{5},\sqrt{5}) \cup (\sqrt{5},\infty)\)