Given the function \( f(x)=\sqrt[3]{x} \), what is the new function after a horizontal stretch by a factor of 4? Option \#1: \( f(x)=\sqrt[3]{4 x} \) Option \#2: \( f(x)=4 \sqrt[3]{x} \) Option \#3: \( f(x)=\sqrt[3]{0.25 x} \) Option \#4: \( f(x)=0.25 \sqrt[3]{x} \)

To perform a **horizontal stretch** on the function \( f(x) = \sqrt[3]{x} \) by a factor of 4, you need to understand how such transformations affect the function. ### Understanding Horizontal Stretch A horizontal stretch by a factor of \( k \) is achieved by replacing \( x \) with \( \frac{x}{k} \) in the function. Mathematically, this transformation is represented as: \[ f_{\text{new}}(x) = f\left(\frac{x}{k}\right) \] For a **horizontal stretch by a factor of 4**, set \( k = 4 \): \[ f_{\text{new}}(x) = \sqrt[3]{\frac{x}{4}} = \sqrt[3]{0.25x} \] ### Evaluating the Options Let's compare this with the given options: 1. **Option 1:** \( f(x) = \sqrt[3]{4x} \) - This represents a **horizontal compression** by a factor of \( \frac{1}{4} \), not a stretch. 2. **Option 2:** \( f(x) = 4 \sqrt[3]{x} \) - This is a **vertical stretch** by a factor of 4, not a horizontal transformation. 3. **Option 3:** \( f(x) = \sqrt[3]{0.25x} \) - This matches our derived transformation for a horizontal stretch by a factor of 4. 4. **Option 4:** \( f(x) = 0.25 \sqrt[3]{x} \) - This is a **vertical compression** by a factor of 0.25, not a horizontal stretch. **Therefore, the correct transformation is Option 3.** **Answer:** Option #3: \( f(x)=\sqrt[3]{0.25 x} \)