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 $X$.

Short answer

For Schedule X

$f\left(x\right)=\left\{\begin{array}{l}0.10xif0<x\le 8500\\ 850+0.15(x-8500)if8500<x\le 34500\\ 4750+0.25(x-34500)if34500<x\le 83600\\ 17025+0.28(x-83600)if83600<x\le 174400\\ 42449+0.33(x-174400)if174400<x\le 379150\\ 110016.15+0.35(x-379150)ifx>379150\end{array}\right.$

Step-by-step solution

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 localid="1645865733938" $0<x\le 8500$.

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

$f\left(x\right)=\left\{\begin{array}{l}0.10xif0<x\le 8500\\ 850+0.15(x-8500)if8500<x\le 34500\\ 4750+0.25(x-34500)if34500<x\le 83600\\ 17025+0.28(x-83600)if83600<x\le 174400\\ 42449+0.33(x-174400)if174400<x\le 379150\\ 110016.15+0.35(x-379150)ifx>379150\end{array}\right.$