Let's start with the initial function [tex]\( f(x) = x^5 \)[/tex] and apply the given transformations step by step.
1. Shift left by 1 unit:
When we shift a function [tex]\( f(x) \)[/tex] to the left by 1 unit, we replace [tex]\( x \)[/tex] with [tex]\( x + 1 \)[/tex]. Thus,
[tex]\[
f(x) = x^5 \quad \Rightarrow \quad f(x+1) = (x+1)^5
\][/tex]
2. Reflect over the [tex]\( y \)[/tex]-axis:
Reflecting a function over the [tex]\( y \)[/tex]-axis involves replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex]. So, we apply this to the shifted function:
[tex]\[
f(x+1) = (x+1)^5 \quad \Rightarrow \quad f(-x+1) = (-x+1)^5
\][/tex]
3. Vertically compress by [tex]\(\frac{1}{3}\)[/tex]:
To vertically compress a function by [tex]\(\frac{1}{3}\)[/tex], we multiply the function by [tex]\(\frac{1}{3}\)[/tex]. Thus,
[tex]\[
f(-x+1) = (-x+1)^5 \quad \Rightarrow \quad \frac{1}{3}(-x+1)^5
\][/tex]
So, after applying all the transformations to [tex]\( f(x) = x^5 \)[/tex], the resulting function is
[tex]\[
\boxed{\frac{1}{3}(-x+1)^5}
\][/tex]
Comparing this result to the options provided:
A. [tex]\( f(x) = \left(-\frac{1}{3} x + 1\right)^5 \)[/tex]
B. [tex]\( f(x) = \frac{1}{3}(-x+1)^5 \)[/tex]
C. [tex]\( f(x) = \frac{1}{3}(-x)^5 + 1 \)[/tex]
D. [tex]\( f(x) = \frac{1}{3}(-x-1)^5 \)[/tex]
The correct answer is:
[tex]\[
\boxed{B}
\][/tex]
