Answer
$f(x,y)$ is homogeneous of degree $zero.$
Work Step by Step
Given
$$f(x, y)=\frac{5 x+2 y}{9 x-4 y}$$
Let
\begin{aligned}
f (t x, ty)&=\frac{5t x+2 ty}{9 tx-4 ty}\\
&=\frac{t(5 x+2 y)}{t(9 x-4 y)}\\
&=\frac{5 x+2 y}{9 x-4 y}\\
&=f(x,t)
\end{aligned}
Thus, $f(x,y)$ is homogeneous of degree zero.
Transforming the given function
\begin{aligned}f(x, y)&=\frac{5 \frac{x}{x}+2 \frac{y}{x}}{9\frac{x}{x}-4 \frac{y}{x}}\\
&=\frac{5 +2 \frac{y}{x}}{9 -4 \frac{y}{x}}
\\
\text{Put} \ \ \ v=\frac{y}{x} \ \ \ \text{so, we get}\\
f(x,y)&=\frac{5 +2 v}{9 -4v}
\end{aligned}
