Question

Let f(x) = x+2/x+6 f^-1(-6) =

Answer 1

We can explicitly find the inverse. If $f^{-1}(x)$ is the inverse of $f(x)$, then

$f\left(f^{-1}(x)\right) = \dfrac{f^{-1}(x)+2}{f^{-1}(x)+6} = x$

Solve for the inverse :

$\dfrac{f^{-1}(x) + 2}{f^{-1}(x) + 6} = x$

$\dfrac{f^{-1}(x) + 6 - 4}{f^{-1}(x) + 6} = x$

$1 - \dfrac4{f^{-1}(x) + 6} = x$

$1 - x = \dfrac4{f^{-1}(x) + 6}$

$f^{-1}(x) + 6 = \dfrac4{1-x}$

$\implies f^{-1}(x) = \dfrac4{1-x} - 6$

Then when x = -6, we have

$f^{-1}(-6) = \dfrac4{1-(-6)} - 6 = \dfrac47-6 = \boxed{-\dfrac{38}7}$

Alternatively, we can first solve for x such that $f(x) = -6$. Then taking the inverse of both sides, $x = f^{-1}(-6)$. (The difference in this method is that we don't compute the inverse for all x.)

We have

$\dfrac{x+2}{x+6} = -6$

$x + 2 = -6 (x + 6)$

$x + 2 = -6x - 36$

$7x = -38$

$\implies x = \boxed{-\dfrac{38}7}$