Continuous Tax Functions

One of the disadvantages of using projects in a course is the time it takes to administer and grade. For many faculty members with large class sizes or large amounts of content to cover, it is not feasible to do a project over several weeks. Fear not! You can still incorporate realistic applications in your classes as activities.

I teach in a flipped environment. My students spend a lot of time outside of class watching videos and doing online homework. I use the face to face class time to have students work in groups at the board or on prepared activities. As I work through this semester’s classes, I plan to post these activities on PBL Pathways. So far I have only incorporated two activities…but more are to come along with a dedicated worksheet page with links to all of them.

Keeping in mind that all of my classes include face to face sections and online section, I need to be able to include the online students in the experience that the face to face students have. For my tax activity, they all work through this worksheet in class or online.

  • Tax Functions Worksheet (PDF | DOCX)

Once the worksheet has been completed, I post a debriefing by working through the process. Below I have shown a typical debriefing. This helps the online students (and those absent from class) get the major ideas of what was done in class.

Feel free to use the PDF I have provided above or modify the Word document to suit your needs.


Tax Functions Worksheet

No one likes to pay taxes. It is a fact of life that each of us pay taxes to provide for government services like roads, parks, water service, ect. However, many people have a flawed vision of how the tax they pay changes as their taxable incomes increases.

For instance, one common notion is that the tax you pay jumps when you move to a new tax bracket. In effect, this notion indicates that a small increase in income (like $1) could lead to a huge increase in tax. An example of a function like this is shown below.

In this graph, if your taxable income increases from $10,000 to $10,001, the tax changes from $100 to $150. Yikes!

This graph is an example of a discontinuous function. The discontinuity, at x = 10,000, indicates that this function may not be drawn without lifting a writing device to jump the gap in the graph.

Luckily, tax functions do not behave this way. A typical tax function might look like the one in the graph below.

This is an example of a continuous function. This function may be drawn without having to lift a writing implement from paper. The slope from 0 to 10,000 is 0.01 indicating that the tax rate is 1%. After 10,000, the graph gets slightly steeper. This slope is 0.015 indicating that the tax rate is slightly higher, 1.5%. Higher tax rates for higher incomes is typical in a graduated tax system.

Piecewise linear functions are typically used to model taxes as a function of taxable income.  For the graduated taxes in the graph above, you might be tempted to write

$latex \displaystyle T\left( x \right)=\left\{ \begin{array}{*{35}{l}}
0.01x \\
0.015x \\
\end{array}\quad \begin{array}{*{35}{l}}
\text{if }0\le x\le 10,000 \\
\text{if }x>10,000 \\
\end{array} \right.$

Which of the graphs below matches this function?

To check the continuity of T(x), we start by computing the two-sided limit $latex \displaystyle \underset{x\to 10,000}{\mathop{\lim }}\,T(x)$. Let’s look at the limit from the left side of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,0.01x=100$

The limit from the right of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,0.015x=150$

Since the one-sided limits at x = 10,000 do not match, the two-sided limit does not exist. Looking at the values of each limit, this matches the right hand graph above.

The continuous tax function has the model

$latex \displaystyle T(x)=\left\{ \begin{array}{*{35}{l}}
0.01x \\
0.015x-50 \\
\end{array} \right.\quad \begin{array}{*{35}{l}}
\text{if }0\le x\le 10,000 \\
\text{if }x>10,000 \\

Now when we compute the one-sided limits, they match. The limit from the left side of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{-}}}{\mathop{\lim }}\,0.01x=100$

The limit from the right of 10,000:

$latex \displaystyle \underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,T(x)=\underset{x\to 10,{{000}^{+}}}{\mathop{\lim }}\,0.015x-50=100$

The value of the function at x = 10000 is

$latex \displaystyle T(10,000)=0.01\left( 10,000 \right)=100$

Since the value of the one-sided limits and the value of the function all match, the function is continuous at x = 10,000. So, the function corresponds to the left graph above. By checking these values, we are checking the definition of continuity at x = 10,000,

$latex \displaystyle \underset{x\to 10,000}{\mathop{\lim }}\,T(x)=T(10,000)$

When checking continuity, we need to check each of these values.