Step 1: Write out the given functions
[tex]f(x)=\frac{9}{x},g(x)=\frac{4}{x+2}[/tex]
Step 2: Find 1/f and g/f
[tex]\frac{1}{f}(x)=\frac{x}{9}[/tex]
Therefore,
[tex]\frac{g}{f}(x)=\frac{4}{x+2}\times\frac{x}{9}=\frac{4x}{9(x+2)}[/tex]
The domain of f is (-∞, ∞) \ {0}.
The domain of g is (-∞, ∞)
If x + 2 = 0, then x = -2.
Therefore g/f is undefined at x = -2.
Hence, the domain of g/f is (-∞, ∞) \ {-2,0}.
Therefore, the domain of g/f is (-∞, -2) ∪ (-2, 0) ∪ (0, ∞)
