To determine the resulting function after applying the given sequence of transformations to [tex]\( f(x) = x^5 \)[/tex], follow these steps:
1. Reflect over the y-axis:
The reflection over the y-axis changes the sign of the input variable [tex]\(x\)[/tex]. Therefore, the function transforms into:
[tex]\[
g(x) = (-x)^5
\][/tex]
2. Vertically compress by [tex]\(\frac{1}{3}\)[/tex]:
A vertical compression by a factor of [tex]\(\frac{1}{3}\)[/tex] scales the function value by [tex]\(\frac{1}{3}\)[/tex]. Thus, the function becomes:
[tex]\[
h(x) = \frac{1}{3} \cdot (-x)^5
\][/tex]
3. Shift left by 1 unit:
Shifting a function to the left by 1 unit means replacing [tex]\(x\)[/tex] with [tex]\(x + 1\)[/tex]. Applying this to the function, we get:
[tex]\[
j(x) = \frac{1}{3} \cdot (-(x + 1))^5
\][/tex]
Simplifying the expression, we have:
[tex]\[
j(x) = \frac{1}{3} \cdot (-x - 1)^5
\][/tex]
After applying all these transformations, the resultant function is:
[tex]\[
f(x) = \frac{1}{3}(-x-1)^5
\][/tex]
Therefore, the correct answer is:
C. [tex]\( f(x) = \frac{1}{3}(-x-1)^5 \)[/tex]
