Let's follow the sequence of transformations one by one to obtain the final function starting from [tex]\( f(x) = x^5 \)[/tex]:
1. Stretch vertically by 3:
- Multiplying the function by 3, we get [tex]\( g(x) = 3x^5 \)[/tex].
2. Reflect across the [tex]\( x \)[/tex]-axis:
- Reflecting the function across the [tex]\( x \)[/tex]-axis means multiplying by -1. Therefore, [tex]\( h(x) = -3x^5 \)[/tex].
3. Shift right 1 unit:
- Shifting the function to the right by 1 unit involves replacing [tex]\( x \)[/tex] with [tex]\( x-1 \)[/tex]. Hence, [tex]\( k(x) = -3(x-1)^5 \)[/tex].
4. Shift up 2 units:
- Shifting the function up by 2 units means adding 2 to the function. Therefore, the final function is [tex]\( p(x) = -3(x-1)^5 + 2 \)[/tex].
Thus, the resulting function after all transformations is:
[tex]\[ f(x) = -3(x-1)^5 + 2 \][/tex]
Comparing with the given choices, the correct answer is:
[tex]\[ \boxed{B. \, f(x) = -3(x-1)^5 + 2} \][/tex]
