Finding inverses of two functions and their compositions to solve for unknown.

Question

$$f(x) = 23x + 27,\;\; g(x) = 12x - d$$

I've found $f^{-1}(x),$ and $\,g^{-1}(x)$, but I don't know how to solve for $d$, given

$$f^{-1}(g^{-1}(x)) = g^{-1}(f^{-1}(x)).$$

How do I do this please?

Answer 1

First, find $f^{-1}(x)$, $g^{-1}(x)$.

$f^{-1}(x):\quad$ Express $x$ as a function of $y$: $y = 23x+ 27\iff \dfrac{y-27}{23} = x$. Switch $x$ and $y$: $$y = \frac{x-27}{23} \implies f^{-1}(x) = \frac{x-27}{23}$$

Similarly found, $g^{-1}(x) = \dfrac{x+d}{12}$.

Now compose $g^{-1}(f^{-1}(x))$ and $f^{-1}(g^{-1}(x))$ and set the two compositions equal to one another (since we are given that the compositions are equivalent.)

Solve for $d$.