To determine the domain and range for the inverse of the function \( f(x) = \frac{1}{4}x + 2 \), we first need to analyze the function itself.
- Determine the domain and range of \( f(x) \):
- The function \( f(x) = \frac{1}{4}x + 2 \) is a linear function.
- The domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \).
- The range of \( f(x) \) is also all real numbers, \( (-\infty, \infty) \).
Since the domain of \( f(x) \) is the range of its inverse \( f^{-1}(x) \), and the range of \( f(x) \) is the domain of its inverse, we conclude:
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Domain of the inverse is the range of \( f(x) \):
- So, the domain of \( f^{-1}(x) \) is \( (-\infty, \infty) \).
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Range of the inverse is the domain of \( f(x) \):
- Therefore, the range of \( f^{-1}(x) \) is \( (-\infty, \infty) \).
Since none of the answer options directly match the conclusion, we can only select one that may represent either the domain or range:
Considering the options:
- \( (-\infty, 0) \): Nope
- \( (2, \infty) \): Nope
- \( (-\infty, 0) \cup (0, \infty) \): Nope
- \( (-\infty, 2) \): Nope
- \( (-\infty, 2) \cup (2, \infty) \): Nope
- \( (0, \infty) \): Nope
Since all given options are false for both domain and range, the correct answer isn't present. However, if we were to only focus on choosing one option, it still wouldn't represent the truth about the inverse's domain or range.
The accurate response:
If strictly picking one option is required despite incorrectness, then it could possibly be assumed as \( (-\infty, 2) \cup (2, \infty) \) but that doesn't accurately reflect the correct outcomes. The correct domain and range of the inverse are both \( (-\infty, \infty) \).