Question

A sales person is paid a weekly salary of $760, plus a commission of 7.5% of all her sales for the week. Express her weekly income in terms of the weekly sales amount. Is this a function? How do you know? What is an appropriate domain?

Answer 1

1. We can see that salesperson's weekly income is the sum of her constant weekly salary ($760) and a commission which is variable and depends on her weekly sales.

So, if we say that y is her weekly income and x is her weekly sales, we can write this as:

y = 760 + 0.075x

Note that we had to change percentage to decimal number dividing it by 100.

2 Since for each value of x there is only one corresponding value of y, we can say that this is a function. For any value of x we input there is only one solution we get - that is the main feature of function and a way to tell if something is really a function.

Since this is a function, it can also be written as:

f(x) = 760 + 0.075x

3. Domain of a function is, basically, set of all values of x for which the function can work. That practically means that, since x is weekly sale, it can not be negative (one cannot make -$500 sale, for example). However, it is possible that she doesn't make a sale one week, making it possible for x to be 0. Also, the value of her sales doesn't have to be integer (it is quite possible that she makes $673.50 sale).

All this means that appropriate domain for this function are positive real numbers including 0.