Question

find the inverse function.
$f(x) = -12\frac{1}{x}$
$f^{-1}(x) = ?\frac{1}{\square}$

Explanation:

Step1: Let \( y = f(x) \)

We start by letting \( y = -12\frac{1}{x} \), which can also be written as \( y = -\frac{12}{x} \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap \( x \) and \( y \) in the equation. So we get \( x = -\frac{12}{y} \).

Step3: Solve for \( y \)

First, we can multiply both sides of the equation \( x = -\frac{12}{y} \) by \( y \) to get \( xy = - 12 \). Then, we divide both sides by \( x \) (assuming \( x
eq0 \)) to solve for \( y \). So \( y=-\frac{12}{x} \). But we can rewrite this as \( y = - 12\frac{1}{x} \), which means the inverse function \( f^{-1}(x)=-12\frac{1}{x} \). Comparing with the given form \( f^{-1}(x)=[\?]\frac{1}{[\ ]} \), the first box (the coefficient) is \(- 12\) and the second box (the denominator) is \(x\). But since the problem seems to have a typo in the original function (maybe it's \( f(x)=-12\frac{1}{x} \) and we need to present the inverse in the form \( f^{-1}(x)=[\?]\frac{1}{[\ ]} \)), from our solution \( y = -\frac{12}{x}=-12\frac{1}{x} \), so the coefficient (the first box) is \(-12\) and the denominator (the second box) is \(x\). But if we consider the form given \( f^{-1}(x)=[\?]\frac{1}{[\ ]} \), we can see that our inverse function \( f^{-1}(x)=-12\frac{1}{x} \), so the first blank is \(-12\) and the second blank is \(x\). But let's re - check the steps.

Wait, maybe the original function is \( f(x)=-12\frac{1}{x}=-\frac{12}{x} \). Let's do the inverse again.

Let \( y = f(x)=-\frac{12}{x} \)

Swap \( x \) and \( y \): \( x = -\frac{12}{y} \)

Solve for \( y \):

Multiply both sides by \( y \): \( xy=-12 \)

Divide both sides by \( x \) (\( x
eq0 \)): \( y = -\frac{12}{x}=-12\frac{1}{x} \)

So the inverse function \( f^{-1}(x)=-12\frac{1}{x} \). So in the form \( f^{-1}(x)=[\?]\frac{1}{[\ ]} \), the first box (the coefficient) is \(-12\) and the second box (the denominator) is \(x\). But if we consider the problem's given form, maybe there is a mis - typing, but following the mathematical process, the inverse function is \( f^{-1}(x)=-12\frac{1}{x} \), so the first blank is \(-12\) and the second blank is \(x\). But since the problem asks for the first box (the coefficient) and the second box (the denominator) in the form \( f^{-1}(x)=[\?]\frac{1}{[\ ]} \), from \( f^{-1}(x)=-12\frac{1}{x} \), the first box is \(-12\) and the second box is \(x\).

Answer:

The first blank (the coefficient) is \(-12\) and the second blank (the denominator) is \(x\). So \( f^{-1}(x)=-12\frac{1}{x} \), so the first box is \(-12\) and the second box is \(x\). If we just consider the first box (the coefficient) as asked in the problem's green box, the answer for the green box is \(-12\).