Question 1 / Vraag 1
Income Tax: Rates of Tax for Individuals / Inkomstebelasting: Belastingkoerse vir individue
Search on the internet (SARS website) for the rates of tax for individuals, for the 2026 tax year
(1 March 2025 - 28 February 2026). This information can be used to write income tax ( as a function
of taxable income . Use this information to create a piecewise defined function , using one
independent variable , to model the rates of tax for individuals. / Gaan soek op die internet
(SARS/SAID webtuiste) vir die belastingkoerse vir individue, vir die 2026 belastingjaar (1 Maart 2025 -
28 Februarie 2026). Hierdie inligting kan gebruik word om inkomstebelasting ( ) as ’ n funksie van die
belasbare inkomste te skryf. Gebruik hierdie inligting om ’ stuksgewys gedefinieerde funksie te
ontwikkel, deur een onafhanklike veranderlike te gebruik, om die belastingkoerse vir individue te
modelleer.

Ask by Hampton Evans. in South Africa

Mar 27,2025

Question

Question 1 / Vraag 1
Income Tax: Rates of Tax for Individuals / Inkomstebelasting: Belastingkoerse vir individue
Search on the internet (SARS website) for the rates of tax for individuals, for the 2026 tax year
(1 March 2025 - 28 February 2026). This information can be used to write income tax ( as a function
of taxable income . Use this information to create a piecewise defined function , using one
independent variable , to model the rates of tax for individuals. / Gaan soek op die internet
(SARS/SAID webtuiste) vir die belastingkoerse vir individue, vir die 2026 belastingjaar (1 Maart 2025 -
28 Februarie 2026). Hierdie inligting kan gebruik word om inkomstebelasting ( ) as ’ n funksie van die
belasbare inkomste te skryf. Gebruik hierdie inligting om ’ stuksgewys gedefinieerde funksie te
ontwikkel, deur een onafhanklike veranderlike te gebruik, om die belastingkoerse vir individue te
modelleer.

Ask by Hampton Evans. in South Africa

Mar 27,2025

UpStudy AI · Accepted Answer

We wish to express the income tax payable, $$ T(x) $$, as a function of taxable income, $$ x $$, by using a piecewise definition. For the 2026 tax year (1 March 2025 – 28 February 2026), SARS publishes a table of tax brackets and rates for individuals. (Note that the figures below are provided as an illustrative example based on recent tax tables; if you are required to use the exact values from the SARS website, you should replace these numbers accordingly.) Assume the SARS table for 2026 is given by the following brackets (with income expressed in South African Rands): 1. For $$ 0 \le x \le 237\,100 $$, the tax rate is 18%. 2. For $$ 237\,100 < x \le 370\,500 $$, the tax is a fixed amount of $$ 42\,678 $$ plus 26% of the excess over $$ 237\,100 $$. 3. For $$ 370\,500 < x \le 512\,800 $$, the tax is $$ 77\,362 $$ plus 31% of the excess over $$ 370\,500 $$. 4. For $$ 512\,800 < x \le 673\,000 $$, the tax is $$ 121\,475 $$ plus 36% of the excess over $$ 512\,800 $$. 5. For $$ 673\,000 < x \le 857\,900 $$, the tax is $$ 179\,147 $$ plus 39% of the excess over $$ 673\,000 $$. 6. For $$ x > 857\,900 $$, the tax is $$ 251\,258 $$ plus 41% of the excess over $$ 857\,900 $$. Using this information, we may write the piecewise defined function as $$ T(x)= \begin{cases} 0.18\,x, & \text{if } 0 \le x \le 237\,100, \[1mm] 42\,678 + 0.26\,(x - 237\,100), & \text{if } 237\,100 < x \le 370\,500, \[1mm] 77\,362 + 0.31\,(x - 370\,500), & \text{if } 370\,500 < x \le 512\,800, \[1mm] 121\,475 + 0.36\,(x - 512\,800), & \text{if } 512\,800 < x \le 673\,000, \[1mm] 179\,147 + 0.39\,(x - 673\,000), & \text{if } 673\,000 < x \le 857\,900, \[1mm] 251\,258 + 0.41\,(x - 857\,900), & \text{if } x > 857\,900. \end{cases} $$ Let’s explain each part step by step: 1. **First bracket $$\left(0 \le x \le 237\,100\right)$$:** Every rand earned is taxed at 18%, so $$ T(x)=0.18\,x. $$ 2. **Second bracket $$\left(237\,100 < x \le 370\,500\right)$$:** The tax up to $$ x=237\,100 $$ (the maximum of the previous bracket) is a fixed amount of $$ 0.18 \times 237\,100 = 42\,678. $$ Any income above $$ 237\,100 $$ is taxed at 26%; thus, if $$ x $$ is in this bracket $$ T(x)=42\,678 + 0.26\,(x-237\,100). $$ 3. **Third bracket $$\left(370\,500 < x \le 512\,800\right)$$:** The cumulative tax up to $$ x=370\,500 $$ is given as $$ 77\,362 $$ (this number results from adding the fixed amount for the first bracket and the tax on income in the second bracket). The remaining income is taxed at 31%: $$ T(x)=77\,362 + 0.31\,(x-370\,500). $$ 4. **Fourth bracket $$\left(512\,800 < x \le 673\,000\right)$$:** Up to $$ x=512\,800 $$ the cumulative tax is $$ 121\,475 $$. Income above that threshold is taxed at 36%: $$ T(x)=121\,475 + 0.36\,(x-512\,800). $$ 5. **Fifth bracket $$\left(673\,000 < x \le 857\,900\right)$$:** Up to $$ x=673\,000 $$ the tax is $$ 179\,147 $$ and income above is taxed at 39%: $$ T(x)=179\,147 + 0.39\,(x-673\,000). $$ 6. **Sixth bracket $$\left(x > 857\,900\right)$$:** Finally, for $$ x $$ above $$ 857\,900 $$, the cumulative tax is $$ 251\,258 $$ plus 41% of the excess over $$ 857\,900 $$: $$ T(x)=251\,258 + 0.41\,(x-857\,900). $$ This piecewise function models the tax payable as a function of taxable income $$ x $$ using the provided rates and fixed amounts for the 2026 tax year. (Remember to verify the actual rates and numbers on the SARS website if precision is required.) The income tax $$ T(x) $$ can be calculated using the following piecewise function based on taxable income $$ x $$: $$ T(x)= \begin{cases} 0.18x, & \text{if } 0 \le x \le 237\,100, \[1mm] 42\,678 + 0.26(x - 237\,100), & \text{if } 237\,100 < x \le 370\,500, \[1mm] 77\,362 + 0.31(x - 370\,500), & \text{if } 370\,500 < x \le 512\,800, \[1mm] 121\,475 + 0.36(x - 512\,800), & \text{if } 512\,800 < x \le 673\,000, \[1mm] 179\,147 + 0.39(x - 673\,000), & \text{if } 673\,000 < x \le 857\,900, \[1mm] 251\,258 + 0.41(x - 857\,900), & \text{if } x > 857\,900. \end{cases} $$ This function defines the tax rate for different income brackets in the 2026 tax year.

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