Income Tax Bands and the Limits of an Inverse
A tax adviser in Bamenda is checking a simplified monthly income-tax schedule used by a small business association. Taxable income is measured in CFA francs and taxed in bands: the first 500,000 FCFA is exempt; the next 500,000 is taxed at 10%; the next 1,000,000 is taxed at 15%; any remaining income is taxed at 25%. The adviser treats tax T(x) as a function of taxable income x, with x ≥ 0. A client with taxable income 1,600,000 FCFA wants to know the tax due. Another client has already paid 260,000 FCFA and wants the corresponding taxable income. The association also wants reassurance that the schedule is consistent at the band boundaries and that the tax function can be reversed to recover income from tax. The adviser must present the piecewise rule, check continuity, and explain any restriction needed before an inverse can be used.
1. Write down T(x) as a piecewise function for x ≥ 0 and use it to calculate the tax on 1,600,000 FCFA.
Read the worked answer
The bands give:
- 0 ≤ x ≤ 500,000: T(x) = 0
- 500,000 < x ≤ 1,000,000: T(x) = 0.10(x − 500,000)
- 1,000,000 < x ≤ 2,000,000: T(x) = 50,000 + 0.15(x − 1,000,000)
- x > 2,000,000: T(x) = 200,000 + 0.25(x − 2,000,000)
Check the boundaries: at x = 1,000,000 the second band gives 0.10(500,000) = 50,000, matching the constant in the third band. At x = 2,000,000 the third band gives 50,000 + 0.15(1,000,000) = 200,000, matching the constant in the fourth band. So the schedule is continuous.
For x = 1,600,000, this lies in the third band: T = 50,000 + 0.15(1,600,000 − 1,000,000) = 50,000 + 0.15(600,000) = 50,000 + 90,000 = 140,000 FCFA.
2. A client has paid 260,000 FCFA. Determine the taxable income and state which band contains it.
Read the worked answer
Since 260,000 > 200,000, the income must lie in the top band, where T(x) = 200,000 + 0.25(x − 2,000,000).
Solve: 200,000 + 0.25(x − 2,000,000) = 260,000 0.25(x − 2,000,000) = 60,000 x − 2,000,000 = 240,000 x = 2,240,000 FCFA.
Check: the tax on 2,240,000 is 200,000 + 0.25(240,000) = 260,000. The answer is consistent with the top band because 2,240,000 > 2,000,000. The taxable income is 2,240,000 FCFA.
3. The association claims that T has an inverse on x ≥ 0. Critically assess this claim and give the inverse formula for T > 200,000 FCFA.
Read the worked answer
The claim is false on the whole domain x ≥ 0 because T(x) = 0 for every x in [0, 500,000]. Many different incomes give the same tax, so T is not one-to-one there and cannot be inverted uniquely. If the domain is restricted to x ≥ 500,000, then T is increasing: its slope is 0.10, then 0.15, then 0.25 in the successive bands, so each positive tax value corresponds to exactly one income.
For T > 200,000, use the top band: T = 200,000 + 0.25(x − 2,000,000) x = 2,000,000 + (T − 200,000)/0.25.
For example, T = 260,000 gives x = 2,000,000 + 60,000/0.25 = 2,240,000 FCFA. For lower positive taxes, use the formula for the band containing the tax value. A tax of 0 cannot be inverted uniquely.