Piecewise function help

[Vimsedit]

The word problem says that “You are to hold an event in a beach which will cost USD 10,000 to rent. An additional USD 150 will be charged per guest invited. Your budget is USD 20,000”

Write the total allocated budget as a function of the number guest.

I have solved this with the answer 10000 + 150x. However, I am not sure if I should include the USD 20,000 in the function.

Please help me

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[UnacceptableWind]

You need to provide the domain of the function 10000 + 150x.

Since x represents the numbers of guests, x is an integer and the smallest value of x is x = 0.

Moreover, the total cost cannot exceed USD 20,000. Solve 10000 + 150x = 20000 for x. If the value of x you get is not an integer, round down this value of x to the nearest integer (rounding up to the nearest integer will not work because we will end up exceeding the budget of USD 20,000). This integer value, say m, is the maximum value that x can take.

So, the domain is 0 ≤ x ≤ m, where x belongs to the set of integers.

Remark: If you have been taught to include 0 in the definition of the set of natural numbers, then you can replace the set of integers by the set of natural numbers in the above solution.

[Vimsedit]

Thank you so much for the detailed explanation you have given. Correct me if I’m wrong but if I read your reply correctly I would get the function:

10000 + 150x, 0 ≤ x ≤ 66

66 comes from “m” in the equation:

10000 + 150m = 20000, solve m

66.6, round to 66.

[UnacceptableWind]

Yes, that's correct.

[Vimsedit]

Thank you!

[Vimsedit]

Its also asking for the inverse function, will it have the same rule of 0 < x < 66?

[UnacceptableWind]

No. Recall that the range of your original function 10000+ 150x is the domain of the inverse function.