Let's solve this step-by-step to determine the function [tex]\( g(x) \)[/tex] after the given transformations on [tex]\( f(x) = x^5 \)[/tex].
### Step 1: Vertical Compression by [tex]\( \frac{1}{2} \)[/tex]
To apply a vertical compression by a factor of [tex]\( \frac{1}{2} \)[/tex], you multiply the entire function by [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ f(x) = x^5 \implies \text{new function} = \frac{1}{2} x^5 \][/tex]
### Step 2: Shift Left by 2 Units
To shift the function horizontally to the left by 2 units, you replace [tex]\( x \)[/tex] with [tex]\( x + 2 \)[/tex]:
[tex]\[ \frac{1}{2} x^5 \implies \frac{1}{2} (x + 2)^5 \][/tex]
### Step 3: Shift Down by 1 Unit
To shift the function vertically downward by 1 unit, you subtract 1 from the entire function:
[tex]\[ \frac{1}{2} (x + 2)^5 \implies \frac{1}{2} (x + 2)^5 - 1 \][/tex]
Thus, the function after applying all the given transformations is:
[tex]\[ g(x) = \frac{1}{2} (x + 2)^5 - 1 \][/tex]
### Conclusion
The correct option, given the sequence of transformations applied to [tex]\( f(x) = x^5 \)[/tex], is:
[tex]\[ \boxed{A. \ g(x) = \frac{1}{2} (x + 2)^5 - 1} \][/tex]
