If \(f(x) = \frac{1}{2 - x}, x \neq 2\), find \(f^{-1}(-\frac{1}{2})\).
Answer Details
To find \(f^{-1}(-\frac{1}{2})\), we need to find the value of x such that \(f(x) = -\frac{1}{2}\).
We know that \(f(x) = \frac{1}{2 - x}\), so we set \(\frac{1}{2 - x} = -\frac{1}{2}\) and solve for x:
\[\frac{1}{2 - x} = -\frac{1}{2}\]
\[2 - x = -2\]
\[x = 4\]
Therefore, \(f^{-1}(-\frac{1}{2}) = 4\).
Note that we were able to solve for \(f^{-1}(y)\) by first setting \(y = f(x)\), then solving for x. This is because the inverse function of f, denoted by \(f^{-1}\), is defined such that \(f^{-1}(f(x)) = x\) for all x in the domain of f.