Question

given the graph of the function $f(x) = \sqrt3{x}$, what is the new function when $f(x)$ is vertically compressed by a factor of $\frac{1}{4}$? option #1 $f(x) = 4\sqrt3{x}$ option #2 $f(x) = \frac{1}{4}\sqrt3{x}$ option #3 $f(x) = \sqrt3{\frac{1}{4}x}$ option #4 $f(x) = \sqrt3{4x}$

Explanation:

Step1: Recall Vertical Compression Rule

For a function \( y = f(x) \), a vertical compression by a factor of \( k \) (where \( 0 < k < 1 \)) transforms the function to \( y = k \cdot f(x) \).

Step2: Apply the Rule to Given Function

The original function is \( f(x)=\sqrt[3]{x} \), and we need a vertical compression by a factor of \( \frac{1}{4} \). Using the vertical compression rule, the new function should be \( f(x)=\frac{1}{4}\cdot\sqrt[3]{x} \), which matches Option #2.

Answer:

Option #2 \( f(x)=\frac{1}{4}\sqrt[3]{x} \)