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Chapter 5: Problem 99

A summer little-league baseball team determines that the cost per player of a group hitting lesson is given by the formula $$ C(x)=\frac{72+2 x}{x} $$ where \(x\) is the number of players in the group and \(C(x)\) is in dollars. a) Determine the cost per player of a group hitting lesson when there are \(2,5,\) and 8 players in the group. b) Find a formula for the inverse of the function and explain what it represents. c) Use the inverse function to determine the number of players in the group lesson when the cost per player is \(\$ 74, \$ 20,\) and \(\$ 11\) (PICTURE CANNOT COPY)

Short Answer

Expert verified
a) Cost: \(38, \)16.4, $11. b) Inverse function: \[C^{-1}(x) = \frac{72}{x - 2}\]. c) Players: 1, 4, 8.

Step by step solution

01

Understand the Given Function

The given function is \[C(x) = \frac{72 + 2x}{x}\]where C(x) represents the cost per player, and x is the number of players.
02

Calculate Cost for Specific Values of x

To find the cost per player for different values of x, substitute x into the function C(x).For x = 2:\[C(2) = \frac{72 + 2(2)}{2} = \frac{72 + 4}{2} = \frac{76}{2} = 38 \text{ dollars}\]For x = 5:\[C(5) = \frac{72 + 2(5)}{5} = \frac{72 + 10}{5} = \frac{82}{5} = 16.4 \text{ dollars}\]For x = 8:\[C(8) = \frac{72 + 2(8)}{8} = \frac{72 + 16}{8} = \frac{88}{8} = 11 \text{ dollars}\]
03

Find the Inverse Function

To find the inverse function, start from the equation \[C(x) = \frac{72 + 2x}{x}\]1. Replace C(x) with y for convenience:\[y = \frac{72 + 2x}{x}\]2. Swap x and y:\[x = \frac{72 + 2y}{y}\]3. Solve for y:\[x = 72/y + 2 \Rightarrow x - 2 = 72/y \Rightarrow y = \frac{72}{x - 2}\]Thus, the inverse function is \[C^{-1}(x) = \frac{72}{x - 2}\].
04

Interpret the Inverse Function

The inverse function \[C^{-1}(x) = \frac{72}{x - 2}\]represents the number of players, x, required for the cost per player to be y dollars.
05

Use the Inverse Function to Determine the Number of Players

Use the inverse function to find the number of players for specific costs per player.For y = 74 dollars:\[x = \frac{72}{74 - 2} = \frac{72}{72} = 1 \text{ player}\]For y = 20 dollars:\[x = \frac{72}{20 - 2} = \frac{72}{18} = 4 \text{ players}\]For y = 11 dollars:\[x = \frac{72}{11 - 2} = \frac{72}{9} = 8 \text{ players}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Function
A cost function helps to determine the cost associated with a certain number of players attending the group hitting lesson. The given cost function is defined as \(C(x) = \frac{72 + 2x}{x}\), where \(C(x)\) represents the cost per player and \(x\) represents the number of players. This function helps to calculate expenses based on different group sizes. For instance, substituting \(x = 2\) gives \(C(2) = 38 \text{ dollars}\). By using this function, we can efficiently manage and predict costs based on group size.
Inverse Function
An inverse function essentially reverses the role of the input and output variables in a function. In this context, the original function is \(C(x) = \frac{72 + 2x}{x}\). To find its inverse, we first replace \(C(x)\) with \(y\), giving \(y = \frac{72 + 2x}{x}\). Swapping \(x\) and \(y\) and solving for \(y\) yields the inverse function \(C^{-1}(x) = \frac{72}{x - 2}\). This inverse function can determine the number of players given a specific cost per player. For example, if the cost per player is \(\$ 74\), then the number of players is 1.
Function Substitution
Function substitution involves plugging specific values into a function to get results. With the cost function \(C(x) = \frac{72 + 2x}{x}\), you can substitute different values of \(x\) to determine the cost per player for those group sizes. For example: \(C(5) = \frac{82}{5} = 16.4 \text{ dollars}\) and \(C(8) = 11 \text{ dollars}\). This technique is vital for precise calculations and better understanding of how changing inputs affect outputs.
Precalculus Problem Solving
Solving precalculus problems often involves understanding and using various mathematical functions. In this example, the concepts of cost function and inverse function are crucial. Finding solutions requires substituting values and algebraic manipulation. Step-by-step, you calculate costs for different values and derive inverse functions to revert costs back to player numbers. This structured approach highlights the importance of methodical problem solving in precalculus.

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Most popular questions from this chapter

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