A drill team is raising money by holding a car wash. The team earns $6.00 for each car washed. The team's expenses include $50.00 for advertising plus $0.50 in materials for each car washed. Let f(x) represent the team's total earning for washing x cars and g(x) represent the team's total expenses for washing x cars. Describe how you can use f(x) and g(x) to obtain a function p(x) that gives the team's profit for washing x cars. Then write a rule for p(x)

The team earns $6.00 for each car washed
The team's expenses include $50.00 for advertising plus $0.50 in materials for each car washed
f(x) represent the team's total earning for washing x cars
g(x) represent the team's total expenses for washing x cars
p(x) that gives the team's profit for washing x cars

We need to describe how can you use f(x) and g(x) to obtain a function p(x) that gives the team's profit for washing x cars, and then write a rule for p(x)

∵ The team earns $6.00 for each car washed

∵ f(x) represent the team's total earning for washing x cars

- Equate f(x) by the product of x and 6

∴ f(x) = 6 x

∵ The team's expenses include $50.00 for advertising plus $0.50

in materials for each car washed

∵ g(x) represent the team's total expenses for washing x cars

- Equate g(x) by the sum of 50 and the product of 0.5 and x

∴ g(x) = 50 + 0.5 x

The profit = total earning - total expenses

∵ f(x) is the total earning for washing x cars

∵ g(x) is the total expenses for washing x cars

∵ p(x) represents the team's profit for washing x cars

- Subtract g(x) from f(x) and equate the difference by p(x)

∴ p(x) = f(x) - g(x)

The function p(x) is obtained by subtracting g(x) from f(x)

∵ f(x) = 6 x

∵ g(x) = 50 + 0.5 x

∴ p(x) = 6 x - (50 + 0.5 x)

- Simplify the right hand side

∴ p(x) = 6 x - 50 - 0.5 x

- Add the like term

∴ p(x) = 5.5 x - 50

The rule of p(x) is p(x) = 5.5 x - 50

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