To find the equation of the new function after performing the given transformations on [tex]\( F(x) = x^5 \)[/tex], let's break it down into clear steps:
### Step 1: Vertically Compress
First, we need to vertically compress the original function [tex]\( F(x) = x^5 \)[/tex] by a factor of [tex]\( \frac{1}{5} \)[/tex].
- Vertically compressing a function by [tex]\( \frac{1}{5} \)[/tex] means multiplying the output of the function by [tex]\( \frac{1}{5} \)[/tex].
So, the vertically compressed function, [tex]\( G(x) \)[/tex], will be:
[tex]\[ G(x) = \frac{1}{5} \cdot x^5 \][/tex]
### Step 2: Horizontally Shift
Next, we need to horizontally shift this new function, [tex]\( G(x) = \frac{1}{5} x^5 \)[/tex], to the right by 3 units.
- Horizontally shifting a function to the right by 3 units involves replacing [tex]\( x \)[/tex] with [tex]\( x - 3 \)[/tex] in the function.
Therefore, the function after a horizontal shift, [tex]\( H(x) \)[/tex], will be:
[tex]\[ H(x) = \frac{1}{5} \cdot (x - 3)^5 \][/tex]
### Conclusion
Combining these steps, the final equation of the new function is:
[tex]\[ H(x) = \frac{1}{5} (x - 3)^5 \][/tex]
Thus, the new function after vertically compressing [tex]\( F(x) = x^5 \)[/tex] by [tex]\( \frac{1}{5} \)[/tex] and then shifting it to the right by 3 units is:
[tex]\[ H(x) = \frac{1}{5} (x - 3)^5 \][/tex]
