Problem
$P(n)$, given below, is the price in dollars for $n$ grams of vitamins. \[ P(n)=0.8 n+5.2 \] Complete the following statements. Let $P^{-1}$ be the inverse function of $P$. Take $x$ to be an output of the function $P$. That is, $x=P(n)$ and $n=P^{-1}(x)$. (a) Which statement best describes $P^{-1}(x)$ ? The amount of vitamins (in grams) for a price of $x$ dollars. The price (in dollars) for $x$ grams of vitamins. The ratio of the price (in dollars) to the number of grams, $x$. The reciprocal of the price (in dollars) for $x$ grams of vitamins. (b) $p^{-1}(x)=$ (c) $P^{-1}(10.4)=\square$
Solution
Step 1 ：The question is asking for the inverse function of $P(n)$, which is denoted as $P^{-1}(x)$. The inverse function is used to find the original input given the output of a function. In this case, $P^{-1}(x)$ would represent the amount of vitamins (in grams) for a price of $x$ dollars.
Step 2 ：The original function is $P(n) = 0.8n + 5.2$. If we replace $P(n)$ with $x$ and $n$ with $P^{-1}(x)$, we get $x = 0.8P^{-1}(x) + 5.2$.
Step 3 ：We can solve this equation for $P^{-1}(x)$ to find the inverse function. The inverse function $P^{-1}(x)$ is $1.25x - 6.5$.
Step 4 ：Finally, we can substitute $x = 10.4$ into the inverse function to find $P^{-1}(10.4)$. The result is $P^{-1}(10.4) = \boxed{6.5}$ grams.
