Question

reflecting the cube root function practice
complete this assessment to review what you’ve learned. it will not count toward your grade.
given the function $f(x) = \sqrt3{x}$, what is the new function when it is reflected over the $y$-axis and horizontally compressed?
option #1: $f(x) = \sqrt3{-3x}$
option #2: $f(x) = \sqrt3{3x}$
option #3: $f(x) = \sqrt3{\frac{1}{3}x}$
option #4: $f(x) = \sqrt3{-\frac{1}{3}x}$
(1 point)

Explanation:

Step1: Reflect over y - axis

For a function \(y = f(x)\), reflecting over the \(y\) - axis gives \(y=f(-x)\). Given \(f(x)=\sqrt[3]{x}\), after reflecting over the \(y\) - axis, the function becomes \(f(x)=\sqrt[3]{-x}\).

Step2: Horizontal compression

A horizontal compression by a factor of \(a>1\) of a function \(y = f(x)\) is given by \(y = f(ax)\). Let's assume the horizontal compression factor is \(3\) (since we need to get a coefficient of \(x\) inside the cube - root that is a non - 1 positive number for compression). So we substitute \(x\) with \(3x\) in the reflected function.
The reflected function is \(y = \sqrt[3]{-x}\), after horizontal compression by a factor of \(3\), we replace \(x\) with \(3x\) in the argument of the cube - root. So the new function is \(y=\sqrt[3]{- 3x}\).

Answer:

Option #1: \(f(x)=\sqrt[3]{-3x}\)