Answer
$f^{-1}(x) = \frac{1+\sqrt {1-20x}}{10} $, where $x\leq -4$
Work Step by Step
We first write,
$x= f(y) =y − 5y^2, x \geq 1\Rightarrow y\leq -4$
Then we solve this equation for $x$ as a function of $y$
$x=y-5y^2$
or
$-x-5y^2+y=0$
or
$5y^2-y+x=0$
For solving $y$ use the Quadratic formula:
$y = \frac{1\pm\sqrt {1-20x}}{10} $
where $x\leq -4$
Since $y\geq 1$ we have the inverse:
$f^{-1}(x) = \frac{1+\sqrt {1-20x}}{10} $
where $x\leq -4$
