Question
Given the function \( f(x)=\sqrt[3]{x} \), what is the new function when it is reflected over the \( y \)-axis, stretching horizontally? Assume that a is a positive integer greater than 1 . Option \#1: \( f(x)=\sqrt[3]{-a x} \) Option \#2: \( f(x)=\sqrt[3]{\frac{1}{-a} x} \) Option \#3: \( f(x)=\sqrt[3]{a x} \) Option \#4: \( f(x)=\sqrt[3]{\frac{1}{a} x} \) (1 point) The best answer is Option \#
Ask by Medina Delgado. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The new function after reflecting over the \( y \)-axis and stretching horizontally by a factor of \( a \) is \( f(x) = \sqrt[3]{\frac{1}{-a} x} \). So, the correct option is **Option #2**.
Solution
The correct transformation involves both reflecting over the \( y \)-axis and stretching horizontally by a factor of \( a \).
1. **Reflection over the \( y \)-axis:** Replace \( x \) with \( -x \), resulting in \( f(-x) = \sqrt[3]{-x} \).
2. **Horizontal Stretch by \( a \):** Replace \( x \) with \( \frac{x}{a} \), resulting in \( f\left(\frac{x}{a}\right) = \sqrt[3]{\frac{x}{a}} \).
Combining these transformations, the new function is:
\[
f\left(-\frac{x}{a}\right) = \sqrt[3]{-\frac{x}{a}} = \sqrt[3]{\frac{-x}{a}}
\]
This corresponds to **Option #2**.
**Answer:** Option #2
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Simplify this solution Mind Expander
To reflect the function \( f(x) = \sqrt[3]{x} \) over the \( y \)-axis, we replace \( x \) with \( -x \), yielding \( f(-x) = \sqrt[3]{-x} \). Next, to stretch the graph horizontally by a factor of \( a \), we replace \( x \) with \( \frac{x}{a} \), resulting in \( f\left(-\frac{x}{a}\right) = \sqrt[3]{-\frac{x}{a}} \). Combining these transformations leads us to the function \( f(x) = \sqrt[3]{-\frac{1}{a}x} \). Thus, the best answer is Option \#2: \( f(x)=\sqrt[3]{\frac{1}{-a} x} \).
