The fictitious state of Aribraska has a graduated state income tax. - Residents pay \( 3 \% \) on the first \( \$ 15,000 \) of income. - The next \( \$ 25,000 \) earned is taxed at a rate of 5\%. - Any money earned above \( \$ 40,000 \) is taxed at 7\%. The income tax for Aribraska is modeled by a piecewise defined function. Over which part of the domain is the piecewise function defined as \[ T(x)=0.05 x-300 ? \] for income \( 0 \leq x \leq 15,000 \) for income \( 15,00040,000 \) DONE

To understand which income values make the expression T(x) = 0.05x – 300 applicable, we first look at the tax structure of Aribraska: 1. For the first $15,000, residents are taxed at 3%. Thus, if x is in the range 0 ≤ x ≤ 15,000, the tax would be 0.03x. 2. For the next $25,000 (i.e., income from $15,000 up to $40,000), the tax rate is 5%. However, one must also include the tax already paid on the first $15,000. The tax on the first $15,000 is 0.03(15,000) = $450. For any income above $15,000, the tax then is 5% of the excess. So, for x in the range 15,000 < x ≤ 40,000, the tax would be:   T(x) = 450 + 0.05(x – 15,000)       = 450 + 0.05x – 750       = 0.05x – 300. 3. For any income above $40,000, additional details apply (taxed at 7% above $40,000) which involve adding up the taxes from the previous segments. Since the expression T(x) = 0.05x – 300 results from including the tax on the first $15,000 and applying 5% on the income from $15,000 to x, it is defined for incomes in the interval:   15,000 < x ≤ 40,000. Thus, the answer is:   for income 15,000 < x ≤ 40,000.