Which statement is true of the function $f(x)=-\sqrt[3]{x}$ ? Select three options.
The function is always increasing.
The function has a domain of all real numbers.
The function has a range of $\{y \mid-\infty< y< \infty\}$.
The function is a reflection of $y=\sqrt[3]{x}$.
The function passes through the point $(3,-27)$.

Step 1 ：First, we analyze the function $f(x) = -\sqrt[3]{x}$.

Step 2 ：Since the cube root function is always increasing, the function $f(x)$ is always decreasing.

Step 3 ：Since the cube root function is defined for all real numbers, the domain of $f(x)$ is all real numbers.

Step 4 ：The range of the cube root function is all real numbers, so the range of $f(x)$ is also all real numbers.

Step 5 ：Since the function $f(x)$ is the negative of the cube root function, it is a reflection of $y = \sqrt[3]{x}$ across the x-axis.

Step 6 ：Finally, we check if the function passes through the point $(3, -27)$. We have $f(3) = -\sqrt[3]{3} = -27$, so the function does pass through the point $(3, -27)$.

Step 7 ：Thus, the correct statements are: \(\boxed{\text{The function is always decreasing, the domain is all real numbers, and the function is a reflection of } y = \sqrt[3]{x} \text{.}}\)