Question

The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars. How do you solve this?

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

See below

Explanation:

All the pieces are continous, since they are polynomials. So, `T(x)` is continuous if it is continous when it switches from a piece to the other. In particular, you need to evaluate two "concurring" pieces at the break point, and see if the result is the same.

So, to answer (i), let's evaluate the first two pieces where `x=6061`:

First piece: `0.1*6061 = 606.1`
Second piece: `606.1 + 1.8(6061-6061)=606.1`

The answer is yes, `T(x)` is continous at `x=6061`

As for (ii), it's the same thing, but with the second and third piece, and `x=32473`.

Second piece: `606.1 + 1.8(32473-6061)=48147.7`
Third piece: `5360.26 + 0.26*(32473-32473)=5360.26`

And thus `T(x)` is discontinous at `x=6061`

This leads to (iii): it is better to gain, for example, `32473$` than `32474$`, because gaining just one more dollar means to pay `5360.26$` taxes instead of `48147.7$`