Suppose daily operating profit, in dollars, for a movie theater is a function of the number of tickets sold with the rule P(x)=8.5x-2,500, for 0≤x≤1,000.
a) Explain why the profit function has an inverse
b) P(400)=900. Rewrite this equation in terms of P^-1.
c) Verify that P^-1(4470)=820 and describe what this equation means in this context.
d) Find a rule for P^-1(x) and then verify the P^-1(P(450))=450
is x the tickets sold? Assuming yes...
a. given a profit, there is only one number of tickets that could have been sold.
b. P^-1(900)=400
c. it means that 4470 of revenue translates to 820 tickets.
P(820)=8.5(820)-2500=4470 check that.
d. I don't understand what the question wants.
a) The profit function has an inverse because it satisfies the conditions necessary for a function to have an inverse. In particular, the given profit function, P(x) = 8.5x - 2,500, is a one-to-one function. This means that each input value corresponds to a unique output value. Therefore, the inverse function exists.
b) To rewrite the equation P(400) = 900 in terms of P^-1, we need to think of P^-1 as a function that "undoes" what the original function does. In other words, it takes the output of the original function and gives us the input that produced that output.
Given P(400) = 900, this means that when 400 tickets are sold, the profit is $900. So, to find the corresponding input value using P^-1, we can plug in 900 as the argument of P^-1:
P^-1(900) = x
c) To verify P^-1(4470) = 820, we can follow the same logic as in part b. According to the equation P^-1(4470) = 820, it means that when the profit is $4,470, the corresponding number of tickets sold is 820. This equation tells us what input value will give us the specified output value in terms of profit.
d) To find a rule for P^-1(x), we can start by setting P(x) equal to a variable, say y:
y = 8.5x - 2,500
Next, we can solve this equation for x to express x in terms of y:
y + 2,500 = 8.5x
x = (y + 2,500) / 8.5
Thus, the rule for the inverse function P^-1(x) is:
P^-1(x) = (x + 2,500) / 8.5
To verify P^-1(P(450)) = 450, we substitute P(450) into P^-1(x):
P^-1(P(450)) = P^-1(8.5 * 450 - 2,500)
= P^-1(3825)
Plugging in the value into the rule for P^-1(x):
P^-1(3825) = (3825 + 2,500) / 8.5
= 4,325 / 8.5
= 509.41
Since the result is not equal to 450, we can conclude that P^-1(P(450)) ≠ 450.