Chapter 2.1, Problem 76E

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

$T(x)=\{\begin{array}{ll}0\hfill & \text{if}0\le x\le 10,000\hfill \\ 0.08x\hfill & \text{if}10,000<x\le 20,000\hfill \\ 1600+0.15x\hfill & \text{if}20,000<x\hfill \end{array}$

1. (a) Find T(5,000), T(12,000), and T(25,000).
2. (b) What do your answers in part (a) represent?

To determine

To find: The value of function $T\left(x\right)$ at given income.

Answer to Problem 76E

Income tax at $T\left(5000\right)$ is zero, at $T\left(12,000\right)$ is $960$, at $T\left(25,000\right)$ is $5350$.

Explanation of Solution

The given function is,

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

Function for income 5,000 is,

$T\left(x\right)=0$

So, income tax at $T\left(5000\right)$ is zero.

Function for income 12,000 is,

$T\left(x\right)=0.08x$

Substitute $12,000$ for x in the above function to find the value of $T\left(12,000\right)$,

$\begin{array}{c}T\left(x\right)=0.08\times 12,000\\ =960\end{array}$

So, income tax at $T\left(12,000\right)$ is $960$

Function for income 20,000 is,

$T\left(x\right)=1600+0.15x$

Substitute $25,000$ for x in the above function to find the value of $T\left(25,000\right)$,

$\begin{array}{c}T\left(25,000\right)=1600+0.15\times 25000\\ =1600+3750\\ =5350\end{array}$

So, income tax at $T\left(25,000\right)$ is $5350$.

Hence, income tax at $T\left(5000\right)$ is zero, at $T\left(12,000\right)$ is $960$, at $T\left(25,000\right)$ is $5350$.

To determine

To Explain: Answer represented in part (a).

Answer to Problem 76E

Income tax is increasing simultaneously with income.

Explanation of Solution

From part (a),

Income tax at $T\left(5000\right)$ is zero, at $T\left(12,000\right)$ is $960$, at $T\left(25,000\right)$ is $5350$.

Income tax is increasing simultaneously with income.