Question

The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars.

Answer 1

T(x) =
(i) Determine whether T is continuous at 6061.
(ii) Determine whether T is continuous at 32,473.
(iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.

Answer 2

T(x) =

Answer 3

Oh, shhiii...take mushrooms lol.
That is a huge function haha

Lets see:
\[T(x)=\left\{\eqalign{
&0.10x; &x\in(0,6061]\\
&606.10+0.18(x-6061); & x\in (6061,32473]\\
&5360.26+0.26(x-32473); &x\in (32473,72784] \\
&15841.12+0.29(x-72784); &x\in (72784, 149897] \\
&38203.89+0.32(x-149897); &x\in (149897,325127]\\
&94277.49+0.36(x-325127); &x\in(325127,\infty) \\
}\right.\]

Answer 4

wow im already confused. -_-

Answer 5

@ganeshie8 may you explain?

Answer 6

That's the same function. lol

Answer 7

okay but what do i do from here

Answer 8

Erm...is this a calculus question?

Answer 9

say, you're working and earning X amount per year

Answer 10

ur salary is X dollars

Answer 11

yea a pre calc question ... & okay

Answer 12

you will have to pay tax on ur salary X,
here, how much % u pay as tax depends on how much u earning

Answer 13

okay!

Answer 14

look at first line,

if u earn \(\le 6061\), then u will have to pay a tax of 10%

Answer 15

I'm leaving this to @ganeshie8 haha

Answer 16

okie.. ive time :) @KeithAfasCalcLover

Answer 17

okay

Answer 18

first understand all the lines,
it simply tells us, how much u have to pay as tax.

next, we can work the discontinuity part

Answer 19

hold on how did you get 10%

Answer 20

0.10x is same as 10% of x

Answer 21

okay so basically all of them are 10 %?

Answer 22

nope,
second line is 18%
third line is 26%
fourth line is 29%
.....
last line is 36%

Answer 23

more u earn, more the government sucks tax money from u

Answer 24

okay and five line is 32 ! gotcha !!

Answer 25

Yup !

lets plot this piecewise function and see, where its continuous and discontinuous

Answer 26

do u have geogebra ?

Answer 27

i used to but i started messing up and i cant re download it idk why?

Answer 28

okie let me plot it and attach a screenshot

Answer 29

thank you

Answer 30

okay, graphing is bit tedious
lets do some algebra quick

Answer 31

(i) Determine whether T is continuous at 6061.

Answer 32

okay lol

Answer 33

\(

T(x)=\left\{\eqalign{
&0.10x; &x\in(0,6061]\\
&606.10+0.18(x-6061); & x\in (6061,32473]\\
&5360.26+0.26(x-32473); &x\in (32473,72784] \\
&15841.12+0.29(x-72784); &x\in (72784, 149897] \\
&38203.89+0.32(x-149897); &x\in (149897,325127]\\
&94277.49+0.36(x-325127); &x\in(325127,\infty) \\
}\right.


\)

Answer 34

when x = 6061,

when u go from left :-
T(x) = .10(6061) = 606.10

when u go from right :-
T(x) = 606.10 + .18(6061-6061) = 606.10+ 0 = 606.10

Answer 35

left side value = right side value

so function is continuous at x = 6061

Answer 36

does that make some sense

Answer 37

made a little sense....

Answer 38

(ii) Determine whether T is continuous at 32,473.

Answer 39

do this same way,
calculate left side & right side values at x = 32473

Answer 40

holdd onn so for (i) i dont get what im suppose to put???

Answer 41

put below :-

"T(x) is continuous at x=6061 because both left side and right side values are same : 606.10 "

Answer 42

??

Answer 43

sorrry okay gotcha ... was confused...

Answer 44

its okay, once u do part ii, u wil get some confidence im sure

Answer 45

See the tax table once

Answer 46

\(

T(x)=\left\{\eqalign{
&0.10x; &x\in(0,6061]\\
&606.10+0.18(x-6061); & x\in (6061,32473]\\
&5360.26+0.26(x-32473); &x\in (32473,72784] \\
&15841.12+0.29(x-72784); &x\in (72784, 149897] \\
&38203.89+0.32(x-149897); &x\in (149897,325127]\\
&94277.49+0.36(x-325127); &x\in(325127,\infty) \\
}\right.


\)

Answer 47

for x = 32473,
left side function is SECOND LINE
right side function is THIRD LINE

Answer 48

so hold on i did .24(32473) = 8442.98 i got ?

Answer 49

evaluate both left side & right side functions for x = 32473, and see if u get the same value

Answer 50

you need to use SECOND LINE for left side function

Answer 51

okay did i do it right for the left side??

Answer 52

no, from where u got .24 ?

Answer 53

left side function :-
606.10+0.18(x−6061)

Answer 54

right side function :-
5360.26+0.26(x−32473)

Answer 55

u just need to put x = 32473 above and see if u get same value

Answer 56

first one i got 5360.26
second one i got 5360.26
so yea its equal..

Answer 57

perfect !
since left side value = right side value, at x = 32473,
T is continuous at x = 32473

Answer 58

getting ?

Answer 59

yea im getting it a little more... but now how do we do (iii)

Answer 60

for iii)
T is continuous at every point, it has no discontinuities. So earning less money for saving tax is not a good idea.

Answer 61

in other words, wid the given piecewise function, since it is continuous, it is never advantageous to earn less money in taxable income

Answer 62

okay thankk you soo muchhh!

Answer 63

np :) iii part is just there to distract you.
It really doesnt help earning less money here. cuz this piecewise function is fully continuous in its range