Question

"P(n), given below, is the price P in dollars for n grams of vitamins.

P(n)=0.2n+ 5.3
Complete the following statements.
Let P^-1 be the inverse function of P.
Take x to be an output of the function P.
(a) Which statement best describes P^-1(x)?
1. The reciprocal of the price in dollars) for x grams of vitamins.
2. The amount of vitamins (in grams) for a price of x dollars.
3. The price in dollars) for x grams of vitamins.
4. The ratio of the price (in dollars) to the number of grams, x.
(b) P^-1 (x) =
(c) P^-1 (6.6) ="

Answer 1

The inverse function P-1(x) represents the amount of vitamins (in grams) for a given price of x dollars. To express P-1(x), we rearrange the original price function P(n)=0.2n + 5.3 to solve for n as P-1(x) = (x - 5.3) / 0.2. For example, P-1(6.6) evaluates to 6.5 grams of vitamins for $6.6.

Step-by-step explanation:

The function P(n), which gives the price P in dollars for n grams of vitamins, is defined as P(n)=0.2n + 5.3. In this context, the inverse function, P-1(x), will undo the operation carried out by P(n).

(a) Statement Best Describing P-1(x)

The correct statement is: 2. The amount of vitamins (in grams) for a price of x dollars. This is because the inverse function will give us the number of grams n for a given price x, essentially reversing the operation done by P(n).

(b) Expression for P-1(x)

To find P-1(x), we need to solve the original equation for n

So, P-1(x) = (x - 5.3) / 0.2.

(c) Calculation of P-1(6.6)

P-1(6.6) = (6.6 - 5.3) / 0.2

= 1.3 / 0.2

= 6.5 grams

The inverse function evaluates to 6.5 when x is 6.6 dollars, meaning that 6.5 grams of vitamins cost $6.6.