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

For a piece-wise function to be continuous we need to only check the function at the nodes i.e. at the starting and end points.

a)

The function T(x) is given by:

T(x)=    0.10 x       if        0<x≤6061

          606.10+0.18(x-6061)  if  6061 <x≤32473

Now to check whether T(x) is continuous at  x=6061 we need to check the left and right hand limit of the function.

Left hand limit at x=6061 is:

lim x→6061    0.10x

      = 6061.10

Also, the right hand limit of function at x=-6061 is:

lim x→6061   606.10+0.18(x-6061)

            = 606.10+.18(6061-6061)

           = 606.10

Hence, the left hand and right hand limit of the function is equal and equal to the value of the function at x=6061

Hence, the function T(x) is continuous at x=6061

b)

Now we have to check that T(x) is continuous at x=32473

The function T(x) is defined by:

  T(x)=   606.10+0.18(x-6061)  if  6061 <x≤32473

 5360.26+0.26(x-32473)     if  32473<x≤72784

Left hand limit at x=32473 is:

lim x→32473   606.10+0.18(x-6061)

= 606.10+0.18(32473-6061)

Hence, left hand limit equal to right hand limit is equal to function value at x=32473

Hence, the function T(x) is continuous at x=32473.

c)

Similarly when we will check at the other nodal points we get that the function is continuous everywhere in the given domain.

the answer to the expression is 83