Question

Income Tax In a certain country, income tax T is assessed according to the following function of income x:

$T\left(x\right)=\left\{\begin{array}{l}[0,\mathrm{if}0\le x\le 10,000],\\ [0.08x,\mathrm{if}10,000<x\le 20,000],\\ [1600+0.15x,\mathrm{if}20,000<x]\end{array}\right\}$

(a) Find$T\left(5,000\right),T\left(12,000\right),\mathrm{and}T\left(25,000\right).$

(b) What do your answers in part (a) represent?

Short answer

(a)

$\begin{array}{l}T\left(5,000\right)=0\\ T\left(12,000\right)=960\\ T\left(25,000\right)=5350\end{array}$

(b)

The results represent the amount of tax paid on incomes of 5000, 12,000, and 25,000.

Step-by-step solution

Part a. Step 1. Use the definition of functions.

A function is defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x, y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part a. Step 2. Analyze the information given.

We are given the function:

$T\left(x\right)=\left\{\begin{array}{l}\left[0,\mathrm{if}0\le x\le 10,000\right],\\ [0.08x,\mathrm{if}10,000<x\le 20,000],\\ [1600+0.15x,\mathrm{if}20,000<x]\end{array}\right\}$

Part a. Step 3. Find values by plugging x-values.

To find T values we proceed as follows:

$\begin{array}{l}T\left(x\right)=\left\{\begin{array}{l}\left[0,\mathrm{if}0\le x\le 10,000\right],\\ [0.08x,\mathrm{if}10,000<x\le 20,000],\\ [1600+0.15x,\mathrm{if}20,000<x]\end{array}\right\}\\ T\left(5,000\right)=0\\ T\left(12,000\right)=0.08\left(12,000\right)=960\\ T\left(25,000\right)=1600+0.15\left(25,000\right)=5350\end{array}$

Part b. Step 1. Use the definition of functions.

A function is defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x, y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Part b. Step 2. Analyze the information given.

We are given the function:

$T\left(x\right)=\left\{\begin{array}{l}\left[0,\mathrm{if}0\le x\le 10,000\right],\\ [0.08x,\mathrm{if}10,000<x\le 20,000],\\ [1600+0.15x,\mathrm{if}20,000<x]\end{array}\right\}$

Part b. Step 3. interpret the results.

$\begin{array}{l}T\left(5,000\right)=0\\ T\left(12,000\right)=960\\ T\left(25,000\right)=5350\end{array}$

The results represent the amount of tax paid on incomes of 5000, 12,000, and 25,000.