Question

The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars.
(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 1

@phi

Answer 2

(i) Determine whether T is continuous at 6061.

that means do you get the same number as x->6061 from the left
and as x->6061 from the right

i.e. as we start at x less than 6061, and let x get bigger
as compared to starting at x more than 6061 and letting x get smaller
does T(x) approach the same number?

Answer 3

To determine a function "f" is continuous at "a" when three conditions are satisfied.
1) "f" is defined at "a"; that is, "a" is in the domain of "f", so that f(a) is a real number.
2)\[\lim_{x \rightarrow a} f(x) \exists.\]
3) \[\lim_{x \rightarrow a} = f(a)\]

Answer 4

yes.
If you want more background, here is a video
http://www.khanacademy.org/math/differential-calculus/limits_topic/continuity-limits/v/limits-to-define-continuity

Answer 5

32473 looks continuous to me
the second line gives the same number as the third line

Answer 6

Yah please allow me to explain..

Answer 7

you should replace x with 32473 in
606.10 + 0.18(x - 6061)

606.10 + 0.18(32473 - 6061)

Answer 8

To find the right-hand limit, look @ values when it close to 32473. We use 3rd line of function.

T(x) = 5360.26 + 0.26(x - 32,473) if 32,473 < x < 72,784.

Answer 9

Well, I will replace 32473 shorty to find right hand limit
5360.26 + 0.26(x - 32,473) = 5360.26 + 0.26(32,473 - 32,473) = 5360.26.

Answer 10

For the right hand limit, we shall use 3rd line because as you can see that
\[32,473 < x \le 72,784\]
because as we approach from the right, the value not equal to 32473.

Answer 11

left-hand limit 606.10 + 0.18(32473 - 6061)=?

Answer 12

no, left hand limit is
606.10 + 0.18(32473 - 6061)
which is not 606.1

Answer 13

We look at values T(x) when "x" is close to 32473, but still less than 32473, we use 2nd line for this one.
\[606.10 + 0.18(x - 6061) = 606.10 + 0.18(6061 - 6061) = 606.1 if 6061 < x \le 32,473\]

Answer 14

@jim_thompson5910

Answer 15

KoreanGirl, which part are you on again?

Answer 16

part (i) or (ii)?

Answer 17

so you just need help with part 3?

Answer 18

Yes, please

Answer 19

ok one sec

Answer 20

Ok it turns out that there are no discontinuities on this graph

Answer 21

If there were, then we might have something like this