Table 6 shows a recent state income tax schedule for...

Table 6 shows a recent state income tax schedule for individuals filing a return in Kansas. $$ \begin{array}{l} \text { SCHEDULE II-SINGLE, HEAD OF HOUSEHOLD, OR } \\ \text { MARRIED FILING SEPARATE } \\ \text { If taxable income is } \\ \text { Over } \quad \text { But Not Over } \quad \text { Tax Due Is } \\ \qquad \begin{array}{rll} \$ 0 & \$ 15,000 & 3.50 \% \text { of taxable income } \\ \$ 15,000 & \$ 30,000 & \$ 525 \text { plus } 6.25 \% \text { of excess over } \$ 15,000 \\ \$ 30,000 & & \$ 1,462.50 \text { plus } 6.45 \% \text { of excess over } \\ & & \$ 30,000 \\ \end{array} \end{array} $$ (A) Write a piecewise definition for the tax due $T(x)$ on an income of $x$ dollars. (B) $\operatorname{Graph} T(x)$. (C) Find the tax due on a taxable income of $\$ 20,000$. Of $\$ 35,000$ (D) Would it be better for a married couple in Kansas with two equal incomes to file jointly or separately? Discuss.

Table 6 shows a recent state income tax schedule for individuals filing a return in Kansas.
$$
\begin{array}{l}
\text { SCHEDULE II-SINGLE, HEAD OF HOUSEHOLD, OR } \\
\text { MARRIED FILING SEPARATE } \\
\text { If taxable income is } \\
\text { Over } \quad \text { But Not Over } \quad \text { Tax Due Is } \\
\qquad \begin{array}{rll}
\$ 0 & \$ 15,000 & 3.50 \% \text { of taxable income } \\
\$ 15,000 & \$ 30,000 & \$ 525 \text { plus } 6.25 \% \text { of excess over } \$ 15,000 \\
\$ 30,000 & & \$ 1,462.50 \text { plus } 6.45 \% \text { of excess over } \\
& & \$ 30,000 \\
\end{array}
\end{array}
$$
(A) Write a piecewise definition for the tax due $T(x)$ on an income of $x$ dollars.
(B) $\operatorname{Graph} T(x)$.
(C) Find the tax due on a taxable income of $\$ 20,000$. Of $\$ 35,000$
(D) Would it be better for a married couple in Kansas with two equal incomes to file jointly or separately? Discuss.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions depending on the interval of the input. It is commonly used to represent situations where a rule changes based on the value of the variable, such as in tax schedules, where different income ranges are subject to different tax rates or formulas.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each segment of the function over its specified interval and carefully indicating the endpoints, continuity, and any jumps. This visual representation helps in understanding how the function behaves across different ranges.

Marginal Tax Rates

Marginal tax rates refer to the tax rate that applies to each additional dollar earned within a specific income bracket. Understanding marginal tax rates is crucial for analyzing tax systems, as the effective rate can change at different levels of income based on the applicable piecewise function segments.

Linear Functions

Linear functions are mathematical expressions with a constant rate of change, represented by a straight line in a graph. In many tax schedules, each segment of the piecewise function is linear, which makes calculations and graphical representations more straightforward.

Filing Status and Joint vs. Separate Filing

The choice between filing jointly or separately can significantly affect tax liabilities due to differences in tax brackets, deductions, and credits. Analyzing separate versus joint filing helps determine the most tax advantageous option for couples, as equal distributions of income may interact differently with the tax schedule based on the filing status.

Transcript

00:01 All right, this is josh, right? and today we're going to have a look at this problem right here.

00:08 So what we have here is actually a tax schedule, an example, tax schedule for kansas state.

00:15 Right.

00:16 So this schedule is for those who are single, head of household, or those who are married, but they would be filing separately.

00:25 So pretty much it's like a single person filing.

00:29 All right.

00:31 Okay, so let's have a look at.

00:32 It all right so right if the single person's taxable income is above zero but only until $15 ,000 in a taxable income then the tax they need to pay would just be 3 .5 % of that particular income now if they go beyond $15 ,000 but they don't go over $30 ,000 what they need to pay now would be $525 ,000 plus 6 .25 % of that which is over $15 ,000.

01:17 All right.

01:18 So that would be, we'll call the first one as category one between zero and 15 ,000.

01:24 And then 15 ,000 or 30 ,000, we'll call that category two.

01:29 And finally we have category three.

01:32 Right.

01:32 So if we have a single person who has a taxable income above $30 ,000.

01:42 All right and the tax they need to pay would be $1 ,462 .50.

01:50 And then they need to pay 6 .45 % of that which is over $30 ,000.

02:02 Okay.

02:03 So now the first thing we're going to do is we're going to set up, right? we're going to set up a piecewise function that would, of course, have this schedule.

02:16 All right.

02:17 So we're going to call that function function t for tax.

02:27 All right.

02:29 So since we have three categories here, all right, we're going to have that one.

02:35 So the first category, right, that would be from zero, right, all the way to 15 ,000.

02:47 Right.

02:47 Category 2 would be from 15 ,000, all the way to 30 ,000.

02:57 And then finally we have category three, right, when taxable income of that individual is above 30 ,000.

03:08 All right.

03:10 Okay, so now let's put in the correct annotations here so that we could be able to calculate the correct tax for each particular category.

03:20 For category one, all right, the individual needs to pay only 3 .5 % of their taxable income.

03:29 So if their taxable income is x, then that means they only need to pay 0 .035 times x.

03:37 So this would be how we compute for their tax due, right, if their taxable income is between 0 to 15 ,000.

03:49 Now, if they go beyond 15 ,000 but not over 30 ,000, so they now fall under category 2.

03:58 So in category two, it says here, they need to pay 525 plus 6 .25 % in excess of 15 ,000.

04:08 All right.

04:08 That means they need to have a base payment of 525.

04:14 And then we need to get the excess of 15 ,000.

04:19 So to get the excess that is simply x minus 15 ,000.

04:25 All right.

04:26 And then whatever the excess is, we need to move.

04:29 Multiply that with 6 .25%.

04:33 All right.

04:34 So now we have the tax due for category two.

04:39 All right.

04:40 Now in category three, according to what we have there, it should have a base amount of 1 ,462 and 50 cents.

04:51 And then anything in excess of 30 ,000, right, should be paid extra at a tax rate of 6.

05:02 Point 45 percent all right so this is now all right this is now our piece -wise function that would represent our tax table right so now let's definitely take let's graph this one and see how it looks like all right so definitely as you go to a higher category right we expect the line to be steeper as the tax rate actually starts to increase okay all right so we're going to start graphing all right tax category one according to what we have it should be 0 .035 that's 3 .5 % of x we're going to replace this with the red okay but then this one should only be from zero all the way to 15 ,000 all right there we go okay, all right.

06:16 Now we go to category two.

06:20 In category two, right, according to our peacewise function, the individual needs to pay a base tax of 525, and then, right, whatever is in excess of 15 ,000, right, they need to pay, all right, a tax rate of 6 .25%.

06:43 All right.

06:45 Change this to blue ink.

06:53 All right.

06:54 But then of course, this one does not begin all the way from zero.

06:57 It actually begins from 15 ,000 all the way to 30 ,000.

07:06 All right.

07:07 It's this blue part of the piecewise function...

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