Question

Piecewise linear functions are often used to model utility or the score between 0 and 100 of a variable.
Consider the income tax that an individual in South Africa had to pay as it was in 2018

Yearly taxable income (R)	Income tax (R)
0 - 0195 850	18% of taxable income above 0
195 851 - 305 850	35 253 + 26% of taxable income above 195 851
305 851 - 423 300	63 853 + 31% of taxable income above 305 851
423 301 - 555 600	100 263 + 36% of taxable income above 423 301
555 601 - 708 310	147 891 + 39% of taxable income above 555 601
708 311 - 1 5000 000	207 448 + 41% of taxable income above 708 311
1 500 001 and above	523 041 + 45% of taxable income above 1 500 001

Let x be the yearly taxable income of an individual in South Africa in Rand
1. Let t be the income tax an individual had to pay in 2018 in rand.
    Express t = f(x) as a function of x and draw its graph. Describe the shape of this graph in mathematical terminology.

Answer 1

The function f(x) can be written as follows, it would be different for different ranges for x as shown below:

x belongs to	f(x)
0 - 195850	0.18*x
195 851 - 305 850	35 253 + (x - 195 851)*0.26
305 851 - 423 300	63853 + (x - 305851)*0.31
423 301 - 555 600	100263 + (x - 423301)*0.36
555 601 - 708 310	147891 + (x - 555601)*0.39
708 311 - 1 5000 000	207448 + (x - 708311)*0.41
1 500 001 and above	523041 + (x - 1500001)*0.45

The graph would be: