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.

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 role="math" localid="1645867218812" $0<x\le 17000$.

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 $17000<x\le 69000$.

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 $69000<x\le 139350$.

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 $139350<x\le 212300$.

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 $212300<x\le 379150$.

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 $x>379150$.

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

$f\left(x\right)=\left\{\begin{array}{l}0.10xif0<x\le 17000\\ 1700+0.15(x-17000)if17000<x\le 69000\\ 9500+0.25(x-69000)if69000<x\le 139350\\ 27087.50+0.28(x-139350)if139350<x\le 212300\\ 47513.50+0.33(x-212300)if212300<x\le 379150\\ 102574+0.35(x-379150)ifx>379150\end{array}\right.$