Which rules apply when building the inverse function (multiple parameters)

Question

I have a function

$f(x) = (3*a)*x-3*b$

I know here the inverse function is

$f^{-1}(y) = (3*a)^{-1} * (y + 3b)$

I don't understand the steps that would lead to this or which math rules I may have forgotten.

Answer 1

Since I ignore your level in mathematics, I'll keep it very simple and not that rigorous.

The function $f$ takes an $x$ and returns a $y$. You want an inverse that takes a $y$ and returns and $x$ in a way that $x, y$ pairs are the same as defined by $f$. So simply call $f:=y$ and clear for $x$:

$$y = 3ax - 3b \implies y+3b=3ax \implies \frac{y+3b}{3a} = x$$

Now we want this to be a function, and we are not used to having functions defined in terms of $y$, so we change the names and say

$$\frac{x+3b}{3a} = y = f^{-1}(x)$$

More briefly,

$$f^{-1}(x)=\frac{x+3b}{3a}$$

which is what you have.