Let's analyze the function [tex]\( f(x) = -\sqrt[3]{x} \)[/tex]:
1. The function is always increasing.
To determine if a function is always increasing, we need to check the derivative:
[tex]\[
\frac{d}{dx}(-\sqrt[3]{x}) = -\frac{1}{3}x^{-\frac{2}{3}}
\][/tex]
For any [tex]\( x \neq 0 \)[/tex], this derivative is negative. This tells us that the function is always decreasing, not increasing. Therefore, this statement is false.
2. The function has a domain of all real numbers.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Since [tex]\( f(x) = -\sqrt[3]{x} \)[/tex] only introduces a negative sign, it does not restrict the domain. Therefore, [tex]\( f(x) \)[/tex] is also defined for all real numbers. This statement is true.
3. The function has a range of [tex]\( \{ y \mid -\infty < y < \infty \} \)[/tex].
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] can output any real number, i.e., its range is all real numbers. Since [tex]\( f(x) = -\sqrt[3]{x} \)[/tex] simply negates these values, it will also cover all real numbers. Therefore, this statement is true.
4. The function is a reflection of [tex]\( y = \sqrt[3]{x} \)[/tex].
The given function [tex]\( f(x) = -\sqrt[3]{x} \)[/tex] is derived by reflecting [tex]\( y = \sqrt[3]{x} \)[/tex] over the x-axis. Therefore, this statement is true.
5. The function passes through the point [tex]\( (3, -27) \)[/tex].
To check this, we calculate [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = -\sqrt[3]{3} = -\sqrt[3]{3} \neq -27.
\][/tex]
Therefore, [tex]\( (3, -27) \)[/tex] is not on the graph of [tex]\( f(x) = -\sqrt[3]{x} \)[/tex]. This statement is false.
Based on this analysis, the true statements are:
- The function has a domain of all real numbers.
- The function has a range of [tex]\( \{ y \mid -\infty < y < \infty \} \)[/tex].
- The function is a reflection of [tex]\( y = \sqrt[3]{x} \)[/tex].
Thus, the correct options are:
[tex]\[ [2, 3, 4] \][/tex]
