Question

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

Explanation:

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

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

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

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

Step3: Solve for \( y \)

First, multiply both sides by \( y \) to get \( xy = - 12 \). Then, divide both sides by \( x \) (assuming \( x
eq0 \)) to solve for \( y \). We have \( y=-\frac{12}{x} \), which can be rewritten in the form \( f^{-1}(x)=- 12\frac{1}{x} \) (or \( f^{-1}(x)=-\frac{12}{x} \)). Comparing with the given form \( f^{-1}(x)=[\?]\frac{1}{[\ ]} \), we can see that the first box (the coefficient) should be \(- 12\) and the second box (the denominator) should be \(x\).

Answer:

The first box is \(-12\) and the second box is \(x\), so \( f^{-1}(x)=-12\frac{1}{x} \) (or in the filled - in form: the first green box is \(-12\) and the white box is \(x\)).