Piecewise Linear Tax Functions
The table below is used to calculate state income tax in Arizona (2011) for taxpayers whose filing status
is single or married filing separate.
Using this table and the taxable income from line 18 of the From 140, a person’s tax may be calculated.
Use this table to answer the questions below.
a. Find the tax if the taxable income is $8000.
Solution This income level falls in the interval over $0, but not over $10,000. In this interval, the tax is
calculated by multiplying the taxable income by 0.0259 to give,
Tax  0.0259 8000  207.20
The tax on $8000 of taxable income is $207.20.
b. Find the tax if the taxable income is $45,000.
Solution A taxable income of $45,000 is over $25,000, but not over $50,000. To calculate the tax, we
must multiply the income by 0.0336 and subtract 149. When we do this, we get
Tax  0.0336  45000   149  1363
The tax on $45,000 of taxable income is $1363.
c.
Let x represent the taxable income in dollars. Find the piecewise linear function T ( x) that
models the tax in dollars.
Piecewise Linear Tax Functions
Solution Looking at parts a and b where specific values of x were supplied, we see that we could use
the formula
0.0259x or 0.0336 x  149 to compute the tax. In fact, we can come up with linear
functions for each range of taxable incomes. Using these formulas, we can write the piecewise
function,
0.0295 x
0.0288 x  29

T ( x)  0.0336 x  149
0.0424 x  589

0.0454 x  1039
if 0  x  10, 000
if 10, 000  x  25, 000
if 25, 000  x  50, 000
if 50, 000  x  150, 000
if x  150, 000
d. Prove that T ( x) is continuous at x  25, 000 .
Solution To prove that T ( x) is continuous at x  25, 000 , we must show that
lim T  x   T  25, 000 
x  25,000
Start by computing the one sided limits corresponding to the two sided limit on the left:
lim T  x  
x  25,000
lim
x  25,000 
 0.0288 x  29 
 0.0288  25, 000   29
 691
lim T  x  
x  25,000
lim
x  25,000 
 0.0336 x  149 
 0.0336  25, 000   149
 691
Since the one sided limits are equal, the two sided limit is equal to the same value,
lim T  x   691
x  25,000
This tells us that the lines corresponding to each piece get closer and closer together as we approach
25,000. If we compute the value of the function at 25,000,
T  25,000   0.0288  25,000   29  691
For taxable incomes closer and closer to $25,000, the tax approaches $691. And at an income of
$25,000, the tax is also $691. This means
T  x  is continuous at x  25, 000 .