Question

cubic and cube root functions and equations
fva 25 - 20: algebra ii q3
which statement is true of the function $f(x) = -\sqrt3{x}$? choose three correct answers.
the function is a reflection of $y = \sqrt3{x}$.
the function has a range of $\\{y \mid \infty < y < \infty\\}$.
the function passes through the point $(3, -27)$.
the function has a domain of all real numbers.
the function is always increasing.

Explanation:

• For the reflection: Multiplying the parent cube root function $y=\sqrt[3]{x}$ by $-1$ reflects it over the x-axis, so $f(x)=-\sqrt[3]{x}$ is this reflection.
• Domain of cube root functions: Cube roots are defined for all real numbers (negative, zero, positive), so the domain is all real numbers.
• Range of cube root functions: As input $x$ takes all real values, $-\sqrt[3]{x}$ also outputs all real values, so the range is all real numbers, written as $\{y \mid -\infty < y < \infty\}$.
• The function is decreasing (not increasing) because the negative coefficient reverses the increasing trend of the parent function.
• For the point $(3,-27)$: $f(3)=-\sqrt[3]{3}

eq -27$, so the function does not pass through this point.

Answer:

1. The function is a reflection of $y = \sqrt[3]{x}$.
2. The function has a range of $\{y \mid -\infty < y < \infty\}$.
3. The function has a domain of all real numbers.