Given the function \( f(x)=\sqrt[3]{x} \), which of the following functions represents \( f(x) \) reflected over the \( x \)-axis and compressed vertically? Option \#1: \( f(x)=\sqrt[3]{-\frac{1}{2} x} \) Option \#2: \( f(x)=-\frac{4}{3} \sqrt[3]{x} \) Option \#3: \( f(x)=-\frac{1}{2} \sqrt[3]{x} \) Option \#4: \( f(x)=\sqrt[3]{-\frac{4}{3} x} \) (1 point) The best answer is Option \#

To reflect the function \( f(x) = \sqrt[3]{x} \) over the \( x \)-axis, we need to negate it, yielding \( -\sqrt[3]{x} \). To compress it vertically, we multiply the function by a factor between 0 and 1—let's take \(-k \sqrt[3]{x}\) where \(k\) is a positive number greater than one. Now, examining the options, Option \#3: \( f(x) = -\frac{1}{2} \sqrt[3]{x} \) effectively negates \( f(x) \) and compresses it vertically by a factor of \( \frac{1}{2} \). Hence, the best answer is Option \#3. To clarify, if we apply a vertical compression factor of 2, we get \( -\frac{1}{2} \sqrt[3]{x} \), perfectly fitting our reflection and compression criteria! Isn't that just a fun twist of mathematical transformations?