Question

suppose that the functions f and g are defined as follows. find g/f. then, give its domain using an interval or Union of intervals. simplify your answers. Edit: please double check the answers.

Answer 1

Step 1: Write out the given functions

$f(x)=\frac{9}{x},g(x)=\frac{4}{x+2}$

Step 2: Find 1/f and g/f

$\frac{1}{f}(x)=\frac{x}{9}$

Therefore,

$\frac{g}{f}(x)=\frac{4}{x+2}\times\frac{x}{9}=\frac{4x}{9(x+2)}$

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, ∞)