To determine which statements are true of the function [tex]\( f(x) = -\sqrt[3]{x} \)[/tex], let's analyze each statement carefully.
### Statement 1: The function is always increasing.
To determine if a function is always increasing, we need to look at the derivative of the function. For [tex]\( f(x) = -\sqrt[3]{x} \)[/tex]:
[tex]\[ f'(x) = -\frac{1}{3} x^{-\frac{2}{3}} \][/tex]
The derivative [tex]\( f'(x) \)[/tex] has the factor [tex]\( -\frac{1}{3} \)[/tex] which is always negative for all [tex]\( x \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] is always decreasing, not increasing. So, this statement is false.
### Statement 2: The function has a domain of all real numbers.
The function [tex]\( f(x) = -\sqrt[3]{x} \)[/tex] involves taking the cube root of [tex]\( x \)[/tex]. The cube root function is defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions on [tex]\( x \)[/tex] because you can take the cube root of any real number.
So, this statement is true.
### Statement 3: The function has a range of [tex]\( \{y \mid -\infty < y < \infty\} \)[/tex].
Since the cube root function, and by extension its negation, can produce any real number as [tex]\( x \)[/tex] ranges over all real numbers, the range of [tex]\( f(x) \)[/tex] is indeed all real numbers. As [tex]\( x \to \infty \)[/tex], [tex]\( -\sqrt[3]{x} \to -\infty \)[/tex], and as [tex]\( x \to -\infty \)[/tex], [tex]\( -\sqrt[3]{x} \to \infty \)[/tex].
So, this statement is true.
### Statement 4: The function is a reflection of [tex]\( y = \sqrt[3]{x} \)[/tex].
The function [tex]\( f(x) = -\sqrt[3]{x} \)[/tex] is the cube root function [tex]\( y = \sqrt[3]{x} \)[/tex] multiplied by [tex]\(-1\)[/tex]. This multiplication reflects the function over the x-axis.
So, this statement is true.
### Statement 5: The function passes through the point [tex]\( (3, -27) \)[/tex].
To check this statement, we substitute [tex]\( x = 3 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3) = -\sqrt[3]{3} \approx -1.4422 \][/tex]
This result is clearly not [tex]\(-27\)[/tex].
So, this statement is false.
Considering the above analysis, the true statements are:
1. The function has a domain of all real numbers.
2. The function has a range of [tex]\( \{y \mid -\infty < y < \infty\} \)[/tex].
3. The function is a reflection of [tex]\( y = \sqrt[3]{x} \)[/tex].
Therefore, the correct selections 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 final answer is:
[tex]\[ [2, 3, 4] \][/tex]
