Question

this question has two parts. first, answer part a. then, answer part b.
part a
lemonade bernardo spent $15 on supplies for his lemonade stand. he charges $1.25 per glass. the function $p(x) = 1.25x - 15$ represents his profit, where $x$ is the number of glasses of lemonade sold.

a. find the inverse function, $p^{-1}(x)$.

• a) $p^{-1}(x) = \frac{x - 15}{1.25}$
• b) $p^{-1}(x) = \frac{x + 15}{1.25}$
• c) $p^{-1}(x) = \frac{x + 15}{2.25}$
• d) $p^{-1}(x) = \frac{2x + 15}{1.25}$

part b

b. what do $x$ and $p^{-1}(x)$ represent in the context of the inverse function?

$x$ represents the select choice and $p^{-1}(x)$ represents the select choice.

c. how many glasses must bernardo sell in order to make $10 in profit?

select choice glasses

Explanation:

Part A

Step1: Start with the profit function

We have the profit function \( P(x) = 1.25x - 15 \). To find the inverse function, we first replace \( P(x) \) with \( y \), so \( y = 1.25x - 15 \).

Step2: Swap x and y

Next, we swap \( x \) and \( y \) to get \( x = 1.25y - 15 \).

Step3: Solve for y

Now, we solve for \( y \). First, add 15 to both sides of the equation: \( x + 15 = 1.25y \). Then, divide both sides by 1.25 to isolate \( y \): \( y=\frac{x + 15}{1.25} \). Since \( y = P^{-1}(x) \), the inverse function is \( P^{-1}(x)=\frac{x + 15}{1.25} \).

In the original function \( P(x)=1.25x - 15 \), \( x \) is the number of glasses sold and \( P(x) \) is the profit. In the inverse function \( P^{-1}(x) \), we have swapped \( x \) and \( P(x) \) from the original function. So in \( P^{-1}(x) \), \( x \) represents the profit, and \( P^{-1}(x) \) represents the number of glasses of lemonade sold.

Step1: Use the profit function

We know the profit function is \( P(x)=1.25x - 15 \), and we want to find \( x \) when \( P(x) = 10 \). So we set up the equation \( 10=1.25x - 15 \).

Step2: Solve for x

First, add 15 to both sides: \( 10 + 15=1.25x \), which simplifies to \( 25 = 1.25x \). Then, divide both sides by 1.25: \( x=\frac{25}{1.25}=20 \).

Answer:

B) \( P^{-1}(x)=\frac{x + 15}{1.25} \)

Part B

b.