To find the domain and range of the inverse of the function \( f(x) = \frac{1}{4x} + 2 \), we first need to identify the domain and range of the original function.
Step 1: Determine the domain of \( f(x) \)
The function \( f(x) = \frac{1}{4x} + 2 \) is defined for all \( x \) except \( x = 0 \), since division by zero is undefined. Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = (-\infty, 0) \cup (0, \infty) \]
Step 2: Determine the range of \( f(x) \)
Next, we need to find the range of \( f(x) \). As \( x \) approaches 0 from the left, \( f(x) \) approaches \( -\infty \). As \( x \) approaches 0 from the right, \( f(x) \) approaches \( +\infty \). The function \( f(x) = \frac{1}{4x} + 2 \) has horizontal asymptotes with behavior that shows \( f(x) \) never actually reaches the value 2. Therefore, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = (-\infty, 2) \cup (2, \infty) \]
Step 3: Determine the domain and range of the inverse function \( f^{-1}(x) \)
The domain of the inverse function \( f^{-1}(x) \) is equal to the range of the original function \( f(x) \), and the range of the inverse function is equal to the domain of the original function:
- Domain of \( f^{-1}(x) \) is \( (-\infty, 2) \cup (2, \infty) \)
- Range of \( f^{-1}(x) \) is \( (-\infty, 0) \cup (0, \infty) \)
Conclusion
Thus, the final answer for the domain and range of the inverse function is:
- Domain: \( (-\infty, 2) \cup (2, \infty) \)
- Range: \( (-\infty, 0) \cup (0, \infty) \)
The response that matches this is:
domain: \( (-\infty, 2) \cup (2, \infty) \) range: \( (-\infty, 0) \cup (0, \infty) \)
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