To reflect the function [tex]\( f(x) = x^3 \)[/tex] over the line [tex]\( y = x \)[/tex], we need to transform the function accordingly.
The correct approach is to switch the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation. Here's the detailed reasoning:
1. Understanding the Reflection: Reflecting a function over the line [tex]\( y = x \)[/tex] involves swapping the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the function [tex]\( y = f(x) \)[/tex].
2. Starting with the Original Function: Given the function [tex]\( f(x) = x^3 \)[/tex], we can write it as:
[tex]\[
y = x^3
\][/tex]
3. Switching Variables: To reflect over the line [tex]\( y = x \)[/tex], we switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation:
[tex]\[
x = y^3
\][/tex]
4. Solving for the New Dependent Variable: The resulting equation can be solved for [tex]\( y \)[/tex] to express it in terms of [tex]\( x \)[/tex]:
[tex]\[
y = \sqrt[3]{x}
\][/tex]
This process transforms the original function to its reflected form. Therefore, the rule that should be applied in this case is:
A. Switch the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation
Hence, the reflection of [tex]\( f(x) = x^3 \)[/tex] over the line [tex]\( y = x \)[/tex] is achieved by switching the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex], confirming choice A as the correct rule to apply.
