Q. 51 - Precalculus Enhanced with Graphing Utilities

Question

Federal Income Tax Two $2011$ Tax Rate Schedules are given in the table below. If $x$ equals taxable income and $y$ equals the tax due, construct a function $y=f\left(x\right)$ for Schedule $X$. 

Step-by-Step Solution

Answer

For Schedule X

$f\left(x\right)=\left\{\begin{array}{l}0.10x if 0<x\le 8500\\ 850+0.15(x-8500) if 8500<x\le 34500\\ 4750+0.25(x-34500) if 34500<x\le 83600\\ 17025+0.28(x-83600) if 83600<x\le 174400\\ 42449+0.33(x-174400) if 174400<x\le 379150\\ 110016.15+0.35(x-379150) if x>379150\end{array}\right.$

Step 1. Form the first function

If the person earns less than or equal to $8500$ dollars then he pays $10$% of the income tax. So it can be written as

$$\begin{gathered} f_1\left(x\right)=10\%·x \\ {f}_1\left(x\right)=0.10x \end{gathered}$$ 

where $0<x\le 8500$.

Step 2. Form the second function

When a person earns more than $8500$ dollars but less than or equal to $34500$ dollars, then he pays a fixed amount $850$ dollars plus  $15$% of everything over $8500$ dollars. So it can be written as

$$\begin{gathered} f_2\left(x\right)=850+15\%·(x-8500) \\ {f}_2\left(x\right)=850+0.15(x-8500) \end{gathered}$$

where $8500<x\le 34500$.

Step 3. Form the third function

When a person earns more than $34500$ dollars but less than or equal to $83600$ dollars, then he pays a fixed amount $4750$ dollars plus  $25$% of everything over $34500$ dollars. So it can be written as

$$\begin{gathered} f_3\left(x\right)=4750+25\%·(x-34500) \\ {f}_3\left(x\right)=4750+0.25(x-34500) \end{gathered}$$

where $34500<x\le 83600$.

Step 4. Form the fourth function

When a person earns more than $83600$ dollars but less than or equal to $174400$ dollars, then he pays a fixed amount $17025$ dollars plus  $28$% of everything over $83600$ dollars. So it can be written as

$$\begin{gathered} f_4\left(x\right)=17025+28\%·(x-83600) \\ {f}_4\left(x\right)=17025+0.28(x-83600) \end{gathered}$$

where $83600<x\le 174400$.

Step 5. Form the fifth function

When a person earns more than $174400$ dollars but less than or equal to $379150$ dollars, then he pays a fixed amount $42449$ dollars plus  $33$% of everything over $174400$ dollars. So it can be written as

$$\begin{gathered} f_5\left(x\right)=42449+33\%·(x-174400) \\ {f}_5\left(x\right)=42449+0.33(x-174400) \end{gathered}$$

where $174400<x\le 379150$.

Step 6. Form the sixth function

When a person earns more than $379150$ dollars then he pays a fixed amount $110016.50$ dollars plus  $35$% of everything over $379150$ dollars. So it can be written as

$$\begin{gathered} f_6\left(x\right)=110016.50+35\%·(x-379150) \\ {f}_6\left(x\right)=110016.50+0.35(x-379150) \end{gathered}$$ 

where $x>379150$.

Step 7. Write the complete function

Using the six functions formed we can write the complete piecewise function as

$f\left(x\right)=\left\{\begin{array}{l}0.10x if 0<x\le 8500\\ 850+0.15(x-8500) if 8500<x\le 34500\\ 4750+0.25(x-34500) if 34500<x\le 83600\\ 17025+0.28(x-83600) if 83600<x\le 174400\\ 42449+0.33(x-174400) if 174400<x\le 379150\\ 110016.15+0.35(x-379150) if x>379150\end{array}\right.$