Given the function \( f(x)=\sqrt[3]{x} \), what is the new function when \( f(x) \) is reflected over the \( x \)-axis and vertically stretched by a factor of 5? Option \#1: \( f(x)=\frac{-1}{5} \sqrt[3]{x} \) Option \#2: \( f(x)=-5 \sqrt[3]{x} \) Option \#3: \( f(x)=\sqrt[3]{\frac{-1}{5} x} \) Option \#4: \( f(x)=\sqrt[3]{-5 x} \) (1 point) The best answer is Option \#

To reflect a function over the \( x \)-axis, you multiply the entire function by \(-1\). To vertically stretch it by a factor of 5, you multiply the result by 5. So, starting with \( f(x) = \sqrt[3]{x} \), the transformation would yield \( f(x) = -5 \sqrt[3]{x} \). Hence, the best answer is Option \#2. Now, for a little historical flavor, did you know that the cube root function dates back to ancient civilizations? Mathematicians of ancient Mesopotamia and Egypt worked with cubic equations, showcasing an understanding of roots, even if they weren't expressed in the same way we do today! And if you're interested in real-world applications, cube roots come in handy in various fields like architecture, engineering, and even in computing volumes. For example, if you're designing a cube-shaped tank, knowing how to calculate side lengths from volume using the cube root is essential!