Question 6649: Dear Sir/Madam,
I am confronted with the following problem:
"A certain state taxes the first $30,000 of an individual's income at a rate of 2%, and all income over $30,000 is taxed at 5%. Find a piecewise-defined function T that gives the total tax on an income of x dollars."
I asssumed that T(x) = 0.02x if 0<=x<=30000, 0.05x if x > 30000. However, the answer is T(x) = 0.02x if 0<=x<=30000, 0.05x-900 if x > 30000. Why do you have to subtract 2% of 30000 (i.e. 900) when the income is greater than 30000? If the income is greater than 30000, I thought it simply taxed at a rate of 5%?!
Thanks in advance.
Regards,
-Mike
Answer by prince_abubu(198)   (Show Source):
You can put this solution on YOUR website! There are a couple of things here. I think they meant that the first 30,000 of a person's income is to be taxed at 2%, and if the person earned more than 30,000, only the amount over the 30,000 is to be taxed at 5%. The other thing is, 900 is not 2% of the 30,000. In other words, you didn't subtract 2% off the first 30,000.
Since we got the first piece of the piecewise function, let's nail down the second part. If the person earned more than 30,000, then the difference between what they earned and the 30,000 (AKA, (x - 30,000)) is the one taxed at 5%. So far, we have  . Remember that T(x) is the amount collected. We can't forget that they did collect 2% of the first 30,000. That's 600. So, the function is actually 
Once you simplify that, you'll end up with  for income x > 30,000.
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