A proposed income tax schedule is shown in the table below.

$\backslash$table$[[\backslash$table$[[$if your income is:$],[$over$]],$but not over,your,$\backslash$table$[[$of the amount$],[$over$]]],[$so$,$$$35,870,-10\%,],[35,104,$$$3,587\mathrm{.}00+23\%,5,870],[,,+27\%,104,660],[17,245,630,+30\%,],[630,1,35,458\mathrm{.}00+39\mathrm{.}5\%,245,630]]$

$($a$)$ Write the piecewise$-$defined function $T$ with input $x$ that models the tax dollars owed as a function of $x,$ the income, with $T(22,000)$$$52,000$$$35,87023\%$$$35,870$$$35,870+$$$1$$$35,87023\%T(x)if0$

$(b)$ Use the function $to$ find $T(22,000).$

$(c)$ Find the tax owed $onan$ income $of$ $$52,000.$

$(d)$ A friend tells you not $to$ earn any money over $$35,870$ because $it$ would raise your tax rate $to23\%on$ all $of$ your income. Test this statement $by$ finding the tax $on$ $$35,870$ and $on$ $$35,870+$$$1$

$(e)$ Does earning more than $$35,870$ raise your income tax rate $to23\%on$ all $of$ your income?

$(f)$ Use a graphing tool $to$ show the graph $ofT(x)$