How do you solve $(f\circ g)(x), f(x)=\frac{5}{x+9}, g(x)=\frac{8}{x}$ to get a domain excluding 0?

Question

I want to find $(f\circ g)(x)$, where $f(x)=\frac{5}{x+9}$, $g(x)=\frac{8}{x}$. I find this to be $\frac{5}{\frac{8}{x}+9}\implies\frac{5x}{9x+8}$.

I want to find the domain next. To do this, I set the denominator equal to zero: $$9x+8=0$$$$\implies x=-\frac{8}{9}$$ So thus, the domain of $(f\circ g)(x)$ excludes $-\frac{8}{9}$. However, if I plug this into wolfram alpha, it says in addition, the domain excludes a zero. How can I arrive at this?

Answer 1

To pass from $\displaystyle \frac{5}{\frac8x + 9}$ to $\displaystyle \frac{5x}{8+9x}$ we need that $x\neq 0$.