Math Snap

STEP 1

Assumptions1. The income tax function $h(x)$ is given by the piecewise function$$h(x)=\left\{\begin{array}{ll} 0.02 x, & \text { if }0 \leq x<6000 \\ 120+0.03(x-6000), & \text { if }6000 \leq x<11,000 \\ 270+0.04(x-11,000), & \text { if } x \geq11,000\end{array}\right.
$$. We are asked to find the values of $h(1260)$, $h(7160)$, and $h(49,070)$.
3. The taxable income is a non-negative real number.

STEP 2

First, let's find the value of $h(1260)$. Since $1260$ is between $0$ and $6000$, we use the first rule of the piecewise function.
$$h(1260) =0.02 \times1260$$

STEP 3

Calculate the value of $h(1260)$.
h(1260) =0.02 \times1260 = \($\)25.20

STEP 4

Now, let's find the value of $h(7160)$. Since $7160$ is between $6000$ and $11000$, we use the second rule of the piecewise function.
$$h(7160) =120 +0.03 \times (7160 -6000)$$

STEP 5

Calculate the value of $h(7160)$.
h(7160) =120 +0.03 \times (7160 -6000) =120 +0.03 \times1160 = \($\)174.80

STEP 6

Finally, let's find the value of $h(49,070)$. Since $49,070$ is greater than $11000$, we use the third rule of the piecewise function.
$$h(49,070) =270 +0.04 \times (49,070 -11,000)$$

Calculate the value of $h(49,070)$.
h(49,070) =270 +0.04 \times (49,070 -11,000) =270 +0.04 \times38,070 = \($\)1,792.80Interpretation$h(1260)$ means that a single person with a taxable income of $1260$ will pay $25.20 in state income tax.
$h(7160)$ means that a single person with a taxable income of $7160$ will pay $174.80 in state income tax.
$h(49,070)$ means that a single person with a taxable income of $49,070$ will pay $1,792.80 in state income tax.