Question

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 $f(x) = 0.05x - 300$?
for income $x > 40,000$
for income $15,000 < x \leq 40,000$
for income $0 \leq x \leq 15,000$
an aribraska resident who earns $32,000 would owe how much in taxes?
$1,740$
$1,300$
$1,600$

Explanation:

Step1: Verify function domain

First, confirm that $x=32,000$ falls in $15,000 < x \leq 40,000$, so use $f(x)=0.05x - 300$.

Step2: Substitute income into function

Plug $x=32000$ into the tax formula.

$$\begin{align*} f(32000)&=0.05\times32000 - 300 \end{align*}$$

Step3: Calculate tax amount

Compute the product then subtract 300.
$0.05\times32000=1600$
$1600 - 300=1300$

Answer:

1. The domain for $f(x)=0.05x - 300$ is: for income $15, 000 < x \leq 40, 000$
2. The tax owed is $\$1,300$