Recall that:
[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)},[/tex]
Now, we know that
[tex]\begin{gathered} (\frac{f}{g})(x)=\frac{x+4}{x^2+2x}, \\ f(x)=\frac{1}{x}\text{.} \end{gathered}[/tex]
Therefore,
[tex]\frac{\frac{1}{x}}{g(x)}=\frac{x+4}{x^2+2x}.[/tex]
Solving the above equation for g(x) we get:
[tex]g(x)=\frac{\frac{1}{x}}{\frac{x+4}{x^2+2x}}.[/tex]
Simplifying the above result we get:
[tex]g(x)=\frac{x^2+2x}{x(x+4)}=\frac{x+2}{x+4}.[/tex]
Answer:
[tex]g(x)=\frac{x+2}{x+4}\text{.}[/tex]
