Question

1. the band at a high school is selling donuts for a local bakery as a fundraiser. the bakery will donate $200 to the band program, plus $3 for each box of donuts that the band sells. the equation that gives the total money, t, raised as a function of the number of boxes sold, n, is $t(n) = 3n + 200$.

a. what is the total money raised if the band sells 84 boxes of donuts?
b. solve for the inverse.
c. what does the inverse tell you?

Explanation:

Part a

Step1: Substitute n=84 into T(n)

$T(84) = 3(84) + 200$

Step2: Calculate multiplication first

$3(84) = 252$

Step3: Add the fixed donation

$T(84) = 252 + 200$

Part b

Step1: Replace T(n) with T

$T = 3n + 200$

Step2: Swap T and n

$n = 3T + 200$

Step3: Isolate the term with T

$n - 200 = 3T$

Step4: Solve for T (inverse function)

$T^{-1}(n) = \frac{n - 200}{3}$

Part c

Step1: Interpret inverse function meaning

The inverse swaps the input and output of the original function. The original function takes number of boxes sold and outputs total money raised, so the inverse does the opposite.

Answer:

a. $\$452$
b. $T^{-1}(n) = \frac{n - 200}{3}$
c. The inverse function tells you the number of boxes of donuts that need to be sold to raise a specific total amount of money.