In a certain country, income tax is assessed as follows....

In a certain country, income tax is assessed as follows. There is no tax on income up to $\$ 10,000 .$ Any income over $\$ 10,000$ is taxed at a rate of $10 \%,$ up to an income of $\$ 20,000$ . Any income over $\$ 20,000$ is taxed at 15$\%$ . (a) Sketch the graph of the tax rate $R$ as a function of the income $I .$ (b) How much tax is assessed on an income of $\$ 14,000 ?$ On $\$ 26,000 ?$ (c) Sketch the graph of the total assessed tax $T$ as a function of the income $I .$

In a certain country, income tax is assessed as follows.
There is no tax on income up to $\$ 10,000 .$ Any income over
$\$ 10,000$ is taxed at a rate of $10 \%,$ up to an income of
$\$ 20,000$ . Any income over $\$ 20,000$ is taxed at 15$\%$ .
(a) Sketch the graph of the tax rate $R$ as a function of the
income $I .$
(b) How much tax is assessed on an income of $\$ 14,000 ?$
On $\$ 26,000 ?$
(c) Sketch the graph of the total assessed tax $T$ as a function
of the income $I .$

Key Concepts

Graphing Piecewise and Step Functions

Graphing piecewise and step functions involves representing each segment of the function on a graph according to its corresponding rule over specific intervals. This visualization captures sudden changes or 'jumps' in the function at threshold points, which is particularly useful in displaying how rates or cumulative figures change in piecewise settings such as tax models.

Marginal Tax Rates

Marginal tax rates refer to the rate of tax applied to the next unit of income earned. In progressive tax systems, income is divided into parts with each portion, above certain thresholds, being taxed at increasingly higher rates, rather than taxing the entire income at a single flat rate.

Progressive Taxation

Progressive taxation is a system where the tax rate increases as the taxable amount increases. This means that lower income levels may be taxed at lower rates or not at all, while higher incomes are subject to higher marginal tax rates. It ensures that taxpayers contribute a fair proportion relative to their earnings.

Piecewise Functions

Piecewise functions are mathematical functions defined by different formulas over different intervals of the input variable. In the context of taxation, they allow us to model scenarios where the tax rate or taxation rule changes at specific income thresholds, reflecting different treatments for different income brackets.

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