I need help answering this problem

Question

A certain state taxes the first $600,000 in property value at a rate of 1%; all value over $600,000 is taxed at 1.25%. Find a piecewise-defined function T that specifies the total tax on a property valued at x dollars. (Simplify your answer completely.)

Answer 1

t(v) = 0.01v for 0 ≤ v ≤ 600,000

t(v) = 6,000 + 0.0125(v - 600,000) for v > 600,000. You are only paying 1.25% for the amount that exceeds $600,000. The first $600,000 will incur $6000 tax.

t is the function

v is the input value in dollars.

Answer 2

Piecewise function basically means having multiple functions where the scenario is different. In this question, there needs to be a scenario where the property value is up to $600,000 (as it states first $600,000 is taxed) at 1%. That means any property value exceeding $600,000 requires a different function. The two functions will be separated in the case that x<=$600,000 and where x>$600,000. For the first function where x<=$600,000, the tax rate is 1% of the property. This can be expressed as "(0.01)x". we multiply by 0.01 to represent the tax paid. In the scenario where x>$600,000, the tax rate has increased to 1.25%. The function will now be expressed as "(0.0125)x. We use 0.0125 to represent the 1.25% tax rate. To make this a piecewise-defined function T we combine both scenarios

T= { (0.01)x, x<=$600,000}

{ (0.0125)x, x>$600,000}