Find the domain of the function. $$ f(x)=\frac{4-x}{x^{2}-9 x} $$

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To find the domain of the function

$$
f(x) = \frac{4 - x}{x^2 - 9x},
$$

we need to determine where the function is defined. The function is undefined wherever the denominator is equal to zero. Therefore, we need to solve the equation:

$$
x^2 - 9x = 0.
$$

We can factor the left-hand side:

$$
x(x - 9) = 0.
$$

Setting each factor equal to zero gives us the solutions:

$$
x = 0 \quad \text{or} \quad x - 9 = 0 \implies x = 9.
$$

Thus, the function $$ f(x) $$ is undefined at $$ x = 0 $$ and $$ x = 9 $$.

The domain of the function is all real numbers except for these points. In interval notation, we can express the domain as:

$$
(-\infty, 0) \cup (0, 9) \cup (9, \infty).
$$

Therefore, the domain of the function $$ f(x) $$ is

$$
\boxed{(-\infty, 0) \cup (0, 9) \cup (9, \infty)}.
$$
The domain of the function is $$(-∞, 0) \cup (0, 9) \cup (9, ∞)$$.

Find the domain of the function.

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