Q. 52 - Precalculus Enhanced with Graphing Utilities

Question

Federal Income Tax Refer to the revised $2011$ tax rate schedules. If $x$ equals taxable income and $y$ equals the tax due, construct a function $y=f\left(x\right)$ for Schedule Y-1. 

Step-by-Step Solution

Answer

For Schedule$Y-1$ ,

$f\left(x\right)=\left\{\begin{array}{l}0.10x if 0<x\le 17000\\ 1700+0.15(x-17000) if 17000<x\le 69000\\ 9500+0.25(x-69000) if 69000<x\le 139350\\ 27087.50+0.28(x-139350) if 139350<x\le 212300\\ 47513.50+0.33(x-212300) if 212300<x\le 379150\\ 102574+0.35(x-379150) if x>379150\end{array}\right.$

Step 1. Form the first function

If they earn less than or equal to $17000$ dollars then they pay $10$% 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 $0<x\le 17000$.

Step 2. Form the second function

When they earn more than $17000$ dollars but less than or equal to $69000$ dollars, then they pays a fixed amount $1700$ dollars plus  $15$% of everything over $17000$ dollars. So it can be written as

$$\begin{gathered} f_2\left(x\right)=1700+15\%(x-17000) \\ {f}_2\left(x\right)=1700+0.15(x-17000) \end{gathered}$$

where $17000<x\le 69000$.

Step 3. Form the third function

When they earn more than $69000$ dollars but less than or equal to $139350$ dollars, then they pays a fixed amount $9500$ dollars plus  $25$% of everything over $69000$ dollars. So it can be written as

$$\begin{gathered} f_3\left(x\right)=9500+25\%(x-69000) \\ {f}_3\left(x\right)=9500+0.25(x-69000) \end{gathered}$$

where $69000<x\le 139350$.

Step 4. Form the fourth function

When they earn more than $139350$ dollars but less than or equal to $212300$ dollars, then they pays a fixed amount $27087.50$ dollars plus  $28$% of everything over $139350$ dollars. So it can be written as

$$\begin{gathered} f_4\left(x\right)=27087.50+28\%(x-139350) \\ {f}_4\left(x\right)=27087.50+0.28(x-139350) \end{gathered}$$

where $139350<x\le 212300$.

Step 5. Form the fifth function

When they earn more than $212300$ dollars but less than or equal to $379150$ dollars, then they pays a fixed amount $47513.50$ dollars plus  $33$% of everything over $212300$ dollars. So it can be written as

$$\begin{gathered} f_5\left(x\right)=47513.50+33\%(x-212300) \\ {f}_5\left(x\right)=47513.50+0.33(x-212300) \end{gathered}$$

where $212300<x\le 379150$.

Step 6. Form the sixth function

When they earn more than $379150$ dollars then they pays a fixed amount $102574$ dollars plus  $35$% of everything over $379150$ dollars. So it can be written as

$$\begin{gathered} f_6\left(x\right)=102574+35\%(x-379150) \\ {f}_6\left(x\right)=102574+0.35(x-379150) \end{gathered}$$

where $x>379150$.

Step 7. Write the complete function

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

$f\left(x\right)=\left\{\begin{array}{l}0.10x if 0<x\le 17000\\ 1700+0.15(x-17000) if 17000<x\le 69000\\ 9500+0.25(x-69000) if 69000<x\le 139350\\ 27087.50+0.28(x-139350) if 139350<x\le 212300\\ 47513.50+0.33(x-212300) if 212300<x\le 379150\\ 102574+0.35(x-379150) if x>379150\end{array}\right.$