Question

You decide to open a café in downtown Mesa. The function f(x)=6500x represents your expected net income (in dollars) after being open x weeks. Before you open, you incur an extra expense of $10,000. What transformation of f is necessary to model this situation? How many weeks will it take to pay off the extra expense?

Answer 1

To model the extra expense of $10,000 for opening a café, subtract it from the income function, resulting in g(x) = 6500x - 10,000. Solving for when g(x) equals zero shows it will take approximately 2 weeks to pay off the expense.

Step-by-step explanation:

The function f(x) = 6500x represents your expected net income in dollars after x weeks. To account for the extra expense of $10,000 before opening, we need to adjust this function by subtracting the amount of this expense:

g(x) = f(x) - 10,000

g(x) = 6500x - 10,000

To determine how many weeks it will take to pay off the extra expense, set g(x) equal to zero (breaking even point) and solve for x:

0 = 6500x - 10,000

6500x = 10,000

x = 10,000 / 6500

x ≈ 1.54

It will take approximately 2 weeks to pay off the extra expense as you cannot have a fraction of a week of operation.