Table 5 shows a recent state income tax schedule for married...

Table 5 shows a recent state income tax schedule for married couples filing a joint return in Kansas. (A) Write a piecewise definition for the tax due $T(x)$ on an income of $x$ dollars. (B) Graph $T(x)$. (C) Find the tax due on a taxable income of 40,000 dollar. Of 70,000 dollar. $$ \begin{array}{l} \text { SCHEDULE I-MARRIED FILING JOINT } \\ \text { If taxable income is } \\ \begin{array}{rll} \text { Over } & \text { But Not Over } & \text { Tax Due Is } \\ \$ 0 & \$ 30,000 & 3.50 \% \text { of taxable income } \\ \$ 30,000 & \$ 60,000 & \$ 1,050 \text { plus } 6.25 \% \text { of excess over } \$ 30,000 \\ \$ 60,000 & & \$ 2,925 \text { plus } 6.45 \% \text { of excess over } \$ 60,000 \end{array} \end{array} $$

Table 5 shows a recent state income tax schedule for married couples filing a joint return in Kansas.
(A) Write a piecewise definition for the tax due $T(x)$ on an income of $x$ dollars.
(B) Graph $T(x)$.
(C) Find the tax due on a taxable income of 40,000 dollar. Of 70,000 dollar.

$$
\begin{array}{l}
\text { SCHEDULE I-MARRIED FILING JOINT } \\
\text { If taxable income is } \\
\begin{array}{rll}

\text { Over } & \text { But Not Over } & \text { Tax Due Is } \\
\$ 0 & \$ 30,000 & 3.50 \% \text { of taxable income } \\
\$ 30,000 & \$ 60,000 & \$ 1,050 \text { plus } 6.25 \% \text { of excess over } \$ 30,000 \\
\$ 60,000 & & \$ 2,925 \text { plus } 6.45 \% \text { of excess over } \$ 60,000
\end{array}
\end{array}
$$

Transcript

00:01 Hello everyone, this is josh.

00:04 Today we're going to answer, right, this question right here.

00:09 So what we have here is actually a tax schedule, right? it's an example for kansas state, right? and this is for those who would be filing or for those who are married and they will be filing jointly.

00:29 So the procedure is not too difficult.

00:32 Just need to determine how much is the taxable income for a particular married couple and from there we need to figure out which category they will fall under all right so we have category one right where in the taxable income is above zero dollars but it does not exceed 30 ,000 dollars so it's exactly 30 ,000 dollars.

00:59 According to the rule okay so whatever the taxable income is just take 3 .5 % of that one okay now what if the combined taxable income goes beyond $30 ,000 but it does not go beyond $60 ,000 that of course allows the couple to fall under category two right where in the tax due right for the couple would be a thousand dollars a thousand fifty dollars and then six point twenty five percent of that which is above thirty thousand dollars all right okay all right now what if the couple's combined taxable income is above sixty thousand dollars okay so if it's above sixty thousand dollars then they fall under category three wherein their tax would be $2 ,925 plus 6 .45 % of the excess amount over $60 ,000.

02:14 All right.

02:17 So the thing here, all right, what we're going to do here is we're going to try to figure out, all right, a piecewise function that would represent this tax.

02:29 Table okay all right so we're going to call that um tax function um t all right t of x now the thing with t of x is that it actually wait actually i think we'll probably put this further down okay i think we'll need um okay this t of x would be in three categories right three categories because one we have a payment which is between zero and 30 ,000 right zero and 30 ,000 dollars right and then we have a second category wherein the amount is 30 ,000 is above 30 ,000, but smaller or equal to 60 ,000.

03:32 And then finally, right, category three, wherein we have the combined taxable income being higher than 60 ,000.

03:45 All right.

03:46 Okay, so now it's time to generate, right, the piecewise parts, all right, or the parts of this piecewise function.

03:55 So let's start with category one.

03:57 So according to category one, if the taxable income is between zero and 30 ,000, all we have to do is just take 3 .5 % of the taxable income.

04:10 Okay, so therefore, for category one, within this taxable income, range, all we have to do is take 3 .5 % of the taxable income.

04:26 So we just have this one.

04:29 0 .35 times x.

04:33 All right.

04:34 So that would be the formula or that would be the part of the piecewise function for that particular part of the domain.

04:42 Okay.

04:43 Now, what about the category two? all right.

04:45 How is that different? okay.

04:46 So according to category two, right, if the taxable income is above 30 ,000 but less than or equal to 60 ,000, then the tax should be 1 ,050 plus 6 .25 % excess of 30 ,000.

05:07 All right.

05:08 So that means in category two, right, the married couple is supposed to pay $1 ,050 already.

05:16 Then whatever is above 30 ,000 right they are supposed to pay oops let's erase this one i think i'll need more space here right they'll need to pay 6 .25 % right oops maybe we'll also need to erase the one below right if you know if the one bill if if if this one is like a particular trend let's erase this too right we'll bring it back later maybe maybe this space is enough x higher than 60 ,000 okay right so notice the difference between the first part and the second part because in the second part you know that we have to add 1 ,050 first right and then whatever is in excess of 30 ,000 so in excess we need to subtract it from 30 ,000 whatever is over we need to multiply that with 6 .25%.

06:47 All right.

06:48 Now in category three, it says here, right, if the taxable income is above 60 ,000, then the couple must pay 2 ,925 and 6 .45 % of excess over 60 ,000.

07:07 So this would be 2 ,925.

07:10 And then the excess, right, for 60 ,000, is going to be, right? we're going to take 6 .45%.

07:28 All right.

07:30 And there we have it.

07:32 So this would be the piecewise function now, right, for the tax schedule, right, of married couple filing jointly, right, at the state of kansas.

07:47 So the next part is we're going to be graphing this one.

07:53 All right.

07:54 Okay, so to graph this one, right, we're going to go here to our graphing calculator.

08:00 Clear this two right here.

08:03 So we're going to call the function as tax, all right? but this would be the first part.

08:09 So we're going to call it x1, right? the first part of the piece.

08:15 All right.

08:15 Oh, you have to adjust also the...

08:19 All right the the x value right because it ranges from zero all the way to past 60 ,000 so maybe let's go with above 70 ,000 let's make this let's just go with zero okay all right so there we go for the y value okay so i think we should start off with zero all right and then the amount is all right so at category three, they need to pay at least 2 ,925.

08:59 So maybe $4 ,000 of tax, you know, might be good enough.

09:05 We'll definitely see if the graph, you know, extends further than this one.

09:10 Let's just make it $6 ,000, i guess, just to be sure.

09:18 Okay, so here we go.

09:20 So for the first part, right, for the first part, for the category one, it says you just need to take 3 .5.

09:28 Percent of a taxable income.

09:33 But this one, it has to be constrained from zero, all the way to 30 ,000.

09:46 All right.

09:47 So notice it cuts here.

09:49 So therefore, for a couple whose taxable income is 30 ,000, they need to pay $1 ,000, $1 ,050.

10:01 All right, okay, great.

10:02 And this $1 ,050 is great because if you remember in category two, minimum that they need to pay is actually $1 ,050.