To find the resulting function after applying the given sequence of transformations to [tex]\( f(x) = x^5 \)[/tex], let's follow each transformation step-by-step:
1. Vertical Compression by [tex]\(\frac{1}{2}\)[/tex]:
When we compress a function vertically by a factor of [tex]\(\frac{1}{2}\)[/tex], we multiply the entire function by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
f(x) = x^5 \implies \frac{1}{2} \cdot x^5
\][/tex]
Therefore, after this transformation, our modified function is:
[tex]\[
g(x) = \frac{1}{2} x^5
\][/tex]
2. Horizontal Shift Left by 2 Units:
To shift the function horizontally to the left by 2 units, we replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex] in the function:
[tex]\[
g(x) = \frac{1}{2}(x + 2)^5
\][/tex]
3. Vertical Shift Down by 1 Unit:
To shift the function vertically down by 1 unit, we subtract 1 from the function:
[tex]\[
g(x) = \frac{1}{2}(x + 2)^5 - 1
\][/tex]
Combining all these transformations, we get the resulting function after applying the sequence of transformations to [tex]\( f(x) = x^5 \)[/tex]:
[tex]\[
g(x) = \frac{1}{2}(x + 2)^5 - 1
\][/tex]
Thus, the correct answer is:
B. [tex]\( g(x) = \frac{1}{2}(x + 2)^5 - 1 \)[/tex]
