### Instruction: Let $f(x) = \displaystyle \frac{1}{ax+b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = 0$. Express your answer in terms of $a$ and/or $b$.### Response: To find the inverse of a function, we switch $x$ and $y$ and then solve for $y$.
So we have $x = \frac{1}{ay+b}$.
To solve for $y$, we multiply both sides by $ay+b$.
That gives $x(ay+b) = 1$.
Distributing the $x$, we have $axy + bx = 1$.
Subtracting $bx$ from both sides, we get $axy = 1 - bx$.
To get $y$ by itself, we divide both sides by $ax$.
That gives $y = \frac{1-bx}{ax}$.
So we want to find all solutions to $f^{-1}(x) = 0$.
So we want to find all solutions to $\frac{1-bx}{ax} = 0$.
Multiplying both sides by $ax$ gives $1-bx = 0$.
So $1 = bx$.
And $bx = 1$ so $x = \frac{1}{b}$.