Answer:
To find the inverse function \(f^{-1}(x)\), we first need to switch the roles of \(x\) and \(f(x)\) and then solve for the new variable \(x\). Let's start with the original function:
\[f(x) = 32x + 132\]
1. Swap \(x\) and \(f(x)\):
\[x = 32f(x) + 132\]
2. Solve for \(f(x)\):
\[32f(x) = x - 132\]
3. Isolate \(f(x)\):
\[f(x) = \frac{x - 132}{32}\]
Now we have the inverse function \(f^{-1}(x)\). To determine its domain and range, we can analyze the properties of this function:
**Domain of \(f^{-1}(x)\):** The domain of \(f^{-1}(x)\) is the set of all possible values that \(x\) can take as an input. In this case, there are no restrictions on the denominator (32), so the domain is all real numbers.
**Range of \(f^{-1}(x)\):** The range of \(f^{-1}(x)\) is the set of all possible values that the function can output. Since there are no restrictions on the numerator (the linear expression \(x - 132\)), the range is also all real numbers.
So, in summary:
**Domain of \(f^{-1}(x)\):** \(-\infty < x < +\infty\)
**Range of \(f^{-1}(x)\):** \(-\infty < f^{-1}(x) < +\infty\)
Explanation: