State the domain and range and explain the importance of the horizontal asymptote in the following case:
The profit, in thousands of dollars, from the sale of x kilograms of tuna fish can be modelled by the function P(x)=(5x-400)/(x+600).
The function for the profit is given as: P(x)=(5x-400)/(x+600)
Note: Here, the domain is x > 0. I have taken this as the domain under the assumption that you cannot sell less than 0 kilograms of tuna and for x = 0 the function is not defined. The range for the domain x> 0 is y<=5.
f(x) = (5x - 400)/ (x + 600)
The horizontal asymptote is f(x) = 5.
We get a horizontal asymptote as even if x--> inf, the value of the function can grow till only 5. So, even if a very large number of tuna is sold the maximum profit is only 5.
The required domain is x>=0 , range is y<=5. The horizontal asymptote places a limit on the maximum profit.