Let's solve the problems step-by-step.
Given function:
[tex]\[ P(n) = 0.4n + 6.6 \][/tex]
We need to determine the inverse function [tex]\(P^{-1}(x)\)[/tex], the meaning of this inverse function, and then evaluate it at 8.8.
### Finding the Inverse Function [tex]\(P^{-1}(x)\)[/tex]:
To find the inverse function, we start with:
[tex]\[ x = P(n) = 0.4n + 6.6 \][/tex]
Solve for [tex]\(n\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ x = 0.4n + 6.6 \][/tex]
Subtract 6.6 from both sides:
[tex]\[ x - 6.6 = 0.4n \][/tex]
Divide by 0.4:
[tex]\[ n = \frac{x - 6.6}{0.4} = 2.5x - 16.5 \][/tex]
Thus, the inverse function is:
[tex]\[ P^{-1}(x) = 2.5x - 16.5 \][/tex]
### Answer Part (a):
To determine which statement best describes [tex]\(P^{-1}(x)\)[/tex], consider the context of the functions:
- [tex]\(P(n)\)[/tex] gives the price in dollars when given the amount of vitamins in grams.
- The inverse function [tex]\(P^{-1}(x)\)[/tex] should therefore give the amount of vitamins in grams when given the price in dollars.
So, the best description for [tex]\(P^{-1}(x)\)[/tex] is:
The amount of vitamins (in grams) for a price of [tex]\(x\)[/tex] dollars.
### Answer Part (b):
The inverse function [tex]\(P^{-1}(x)\)[/tex]:
[tex]\[ P^{-1}(x) = 2.5x - 16.5 \][/tex]
### Answer Part (c):
Evaluate the inverse function at [tex]\(x = 8.8\)[/tex]:
[tex]\[ P^{-1}(8.8) = 2.5 \times 8.8 - 16.5 = 5.5 \][/tex]
### Summary:
(a) The best description of [tex]\(P^{-1}(x)\)[/tex] is:
The amount of vitamins (in grams) for a price of [tex]\(x\)[/tex] dollars.
(b) [tex]\( P^{-1}(x) = 2.5x - 16.5 \)[/tex]
(c) [tex]\( P^{-1}(8.8) = 5.5 \)[/tex]
