To reflect the function [tex]\( f(x) = x^3 \)[/tex] over the [tex]\( y \)[/tex]-axis, we need to perform a specific transformation. Let's go through this step by step:
1. Understanding the reflection over the [tex]\( y \)[/tex]-axis:
- Reflecting a graph over the [tex]\( y \)[/tex]-axis means that every point on the graph that was at [tex]\((x, y)\)[/tex] will be moved to [tex]\((-x, y)\)[/tex].
- In terms of the function [tex]\( f(x) \)[/tex], reflecting [tex]\( f(x) \)[/tex] over the [tex]\( y \)[/tex]-axis involves replacing [tex]\( x \)[/tex] with [tex]\(-x \)[/tex].
2. Substituting [tex]\(-x\)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
- Start with the given function [tex]\( f(x) = x^3 \)[/tex].
- To reflect this function over the [tex]\( y \)[/tex]-axis, substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]. This gives us:
[tex]\[
f(-x) = (-x)^3
\][/tex]
3. Simplifying [tex]\( f(-x) \)[/tex]:
- Next, we simplify the expression [tex]\((-x)^3\)[/tex]:
[tex]\[
(-x)^3 = -x^3
\][/tex]
- Thus, the reflected function is [tex]\( f(-x) = -x^3 \)[/tex].
After analyzing the choices, it is clear that:
- Choice B: Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] and simplify is the correct rule to apply for reflecting the function [tex]\( f(x) = x^3 \)[/tex] over the [tex]\( y \)[/tex]-axis.
Therefore, the correct answer is:
B. Substitute [tex]\(-x\)[/tex] for [tex]\( x \)[/tex] and simplify [tex]\( f(-x) \)[/tex].
