Question

Income Tax Suppose that \(I(x)\) represents the income of an individual in year \(x\) before taxes and \(T(x)\) represents the individual's tax bill in year \(x .\) Find a function \(N\) that represents the individual's net income (income after taxes) in year \(x\).

Short answer

The net income function is \(N(x) = I(x) - T(x)\).

Step-by-step solution

Understand the problem

We need to find a function that represents the net income, which is the income after taxes have been subtracted from the total income.

Define the given functions

We are given that \(I(x)\) represents the income before taxes and \(T(x)\) represents the taxes for year \(x\).

Formulate the net income function

Net income can be calculated by subtracting the taxes from the total income. Thus, the function for the net income, \(N(x)\), can be defined as follows: \[ N(x) = I(x) - T(x) \]

Write the final function

The function for the individual's net income in year \(x\) is therefore: \[ N(x) = I(x) - T(x) \]

Key concepts

Income Before Taxes

Income before taxes is the total amount of money someone earns in a year, before any taxes are deducted. Just think of it as the gross income or your salary.
For example, if you have a job that pays you $50,000 per year, that’s your income before taxes. This is the starting point for any tax calculations.
The exercise defines this as the function:
I(x)
In essence, I(x) represents how much money you earn in year x without taking into account any deductions, including taxes.

Tax Calculation

Tax calculation involves determining the amount of money you need to pay to the government based on your income. Not all of the money you earn goes into your pocket; a certain portion is taken out as taxes.
In the exercise, the tax amount is given as:
T(x)
So, T(x) is the function that represents the taxes you owe in year x . It's important to understand that taxes can be calculated using different methods, including flat rates or progressive rates (where higher income brackets are taxed at higher rates).
Understanding how taxes are calculated helps you plan better for financial decisions.

Basic Algebra

Basic algebra helps us manipulate equations and functions to solve for unknowns or express relationships, like we did in our exercise.
In this case, we used basic algebra to find the net income ( N(x)) by subtracting the tax function ( T(x)) from the income function ( I(x)).
This resulted in the net income function:
N(x) = I(x) - T(x)
By understanding and applying basic algebraic operations, we can simplify and solve complex real-world problems like finding net income after taxes.
Remember, algebra is a fundamental tool in mathematics, and it can be applied to a wide range of problems beyond just tax calculations.