Tax notes: Dependent filing requirements

Some clients asked me whether their dependents are required to file tax, I hope the following graph can answer this question clearly.

This graph was created based on the IRS’ “form language” in IRS Publication 929 as reproduced below:

Filing Requirement Worksheet for Most Dependents
1.	Enter dependent's earned income plus $350
2.	Minimum amount	$1,100
3.	Compare lines 1 and 2. Enter the larger amount
4.	Maximum amount	12,550
5.	Compare lines 3 and 4. Enter the smaller amount
6.	Enter the dependent’s gross income. If line 6 is more than line 5, the dependent must file an income tax return. If the dependent is married and his or her spouse itemizes deductions on a separate return, the dependent must file an income tax return if line 6 is $5 or more

The form language can be directly translate to a formula:

$f(x) = {\rm min}({\rm max}(x+350, 1100), 12550)$

(I guess min() and max() are not “true” mathematical functions, while WolframAlpha has it, GeoGebra does not.) I have to translate it to the traditional math function as:

${\rm Gross\_Income}(x)= \left\{ \begin{array}{ll} 1100 & 0\leq x \leq 1100-350 \\ x+350 & 1100-350\leq x\leq 12550-350\\ 12550 & 12550-350\leq x \\ \end{array} \right.$

While the IRS form language uses gross income, it is more insightful to use unearned income (investment) to determine the dependent filing requirement. The formula is:

${\rm Unearned\_Income}(x)= \left\{ \begin{array}{ll} 1100 -x & 0\leq x \leq 1100-350 \\ 350 & 1100-350\leq x\leq 12550-350\\ 12550 -x & 12550-350\leq x \leq 12550 \\ 0 & 12550\leq x \\ \end{array} \right.$

From either the formula or the graph, we can conclude:

when the dependent has zero earned income, she is allowed to have up to $1,100 unearned income without filing tax.
The allowed unearned income reduces linearly to $350 when earned income is between $0 and $750.
The allowed unearned income stay flat when earned income is between $750 and $12,200
The allowed unearned income reduces linearly to $0 when earned income is between $12,200 and $12,550, the standard deduction for single status.

When filing dependent return, be sure to check the “can be claimed as a dependent” box, or it results in a conflict and both the dependent and the person which claimed the dependent will receive an IRS letter. Also when the dependent has earned income, you can open a Roth IRA account for the dependent up to the earned income, or the maximum contribution allowed for the year ($6,000 for 2021), whichever is lower.