A study on how circus ticket pricing influences ticket sales shows that there is an average increase of 4 people for every $3 decrease on the price of the ticket. A circus owner sells an average of 340 tickets when the price of a ticket is $75. If x represents the number of times the price decreases by $3, which function describes the revenue earned by the circus owner in terms of x?

Let the price of a ticket be originally T dollars, and the number of clients be N.

let the price decrease by x 3-dollars.

"there is an average increase of 4 people for every $3 decrease on the price of the ticket."

means:

if the price is decreased by 1-3$:

then N become N+4, and T becomes T-3

if the price is decreased by 2-3$:

then N become N+4+4=N+2*4, and T becomes T-3-3=T-2*3

So if the price is decrease by x-3 dollars:

N becomes N+4x, and T becomes T-3x

"A circus owner sells an average of 340 tickets when the price of a ticket is $75."

In this case N=340, and T=75$

If the owner does not change the price ticket, x=0, the revenue is 340*75,

If the owner decreases the price of the tickets by x-3$, then the revenue will be

(N+4x) (T-3x) = (340+4x) (75-3x) dollars,

If R is the function of the revenue depending on x, then

R (x) = (340+4x) (75-3x) dollars

Answer: R (x) = (340+4x) (75-3x) dollars