A certain state taxes the first $400,000 in property value at a rate of 1%; all value over $400,000 is taxed at 1.75%. Find a piecewise-defined function T that specifies the total tax on a property valued at x dollars. (Simplify your answer completely.)
0.01x if 0<x<400,000
T(x)= 0.0175x-500 if x>400,000
are my numbers right? (By the way the greater/less than signs are equal to as well)
your first part is correct
for the 2nd part, I would have it as
T(x) = .0175x - 3000 , x ≥ 400,000
(so I have taken 1.75% on the whole thing, but then subtracted the 0.75% part for the first 400,000 which is 3000.
Where did you get your $500 from ? )
Yes, your numbers are correct. The piecewise-defined function T you provided specifies the total tax on a property valued at x dollars.
In the first case, if the property value x is between 0 and $400,000 (0 < x < 400,000), the tax rate is 1% or 0.01. So the tax T(x) is simply equal to 0.01 times the property value x: T(x) = 0.01x.
In the second case, if the property value x is greater than $400,000 (x > 400,000), the tax rate increases to 1.75% or 0.0175. However, since the first $400,000 is already taxed at 1%, we need to subtract the tax already paid on that amount (which is $400,000 * 0.01 = $4,000) from the total tax. So the tax T(x) is equal to 0.0175 times the property value x minus $4,000: T(x) = 0.0175x - 4,000.
Therefore, the piecewise-defined function T that specifies the total tax on a property valued at x dollars is as follows:
T(x) =
0.01x if 0 < x < 400,000,
0.0175x - 4,000 if x > 400,000.