Answer
$(a)$
$f(x) = \left\{ \begin{array}{ll} \frac{x}{10} & \quad 0\leq x \leq 20000 \\ \frac{x-20000}{5}+2000 & \quad x > 20000\end{array} \right. $
$(b)$
$f^{-1}(x) = \left\{ \begin{array}{ll} 10x & \quad 0\leq x \leq 2000 \\ 5x+10000 & \quad x > 2000\end{array} \right. $
The inverse function, now $f^{-1}(x)$ is income for given $x$ that is the tax amount paid.
$(c)$ It would require $60,000$ Euros
Work Step by Step
$(a)$ According to the information given, we can write the following function (where $x$ is income and $f(x)$ is tax for given income) :
$f(x) =y= \left\{ \begin{array}{ll} \frac{x}{10} & \quad 0\leq x \leq 20000 \\ \frac{x-20000}{5}+2000 & \quad x > 20000\end{array} \right. $
$(b)$ To find $f^{-1}$, we have to replace $x$ by $y$ and vice versa. For easier calculation, we will do it one by one in the two functions we have.
$1.$
$y=\frac{x}{10}$
$x=10y$
$y=10x$
And the interval will be:
$0\leq 10y \leq 20000$
$0\leq y \leq 2000$
$2.$
$y=\frac{x-20000}{5}+2000$
$x=\frac{y-20000}{5}+2000$
$5x=y-20000+10000$
$y=5x+10000$
And the interval will be:
$5y=x-20000+10000$
$x=5y+10000$
Input the $x$ value in the interval:
$5y+10000\gt 20000$
$5y\gt 10000$
$y\gt 2000$
$f^{-1}(x) = \left\{ \begin{array}{ll} 10x & \quad 0\leq x \leq 2000 \\ 5x+10000 & \quad x > 2000\end{array} \right. $
Note. in the inverse function, now $f^{-1}(x)$ is income for given $x$ that is the tax amount paid.
$(c)$ If the tax paid was $10,000$, we have to use the second function, as $10000$ is $\gt2000$
$y=5\times 10000 +10000=60000$
It would require $60,000$ Euros
