To reflect a function [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex], you need to switch the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation of the function.
Given the function [tex]\( f(x) = x^3 \)[/tex], follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] to write the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = x^3
\][/tex]
2. Switch the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = y^3
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[3]{x} \text{ or } y = x^{1/3}
\][/tex]
Therefore, the rule that should be applied to reflect [tex]\( f(x) = x^3 \)[/tex] over the line [tex]\( y = x \)[/tex] involves switching the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation.
The correct answer is:
B. Switch the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation
