To determine the domain of the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \log_2(x + 2) \), we first need to establish the range of the original function \( f(x) \).
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Finding the domain of \( f(x) \): The function \( f(x) = \log_2(x + 2) \) is defined when the argument of the logarithm is positive: \[ x + 2 > 0 \implies x > -2 \] Therefore, the domain of \( f(x) \) is \( (-2, \infty) \).
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Finding the range of \( f(x) \): As \( x \) approaches \( -2 \), \( f(x) \) approaches \( \log_2(0) \), which approaches \( -\infty \). As \( x \) goes to \( \infty \), \( f(x) \) also approaches \( \infty \). Hence, the range of \( f(x) \) is: \[ (-\infty, \infty) \]
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Finding the domain of \( f^{-1}(x) \): The domain of the inverse function \( f^{-1}(x) \) is equal to the range of the original function \( f(x) \). Since the range of \( f(x) \) is \( (-\infty, \infty) \), the domain of \( f^{-1}(x) \) is also: \[ (-\infty, \infty) \]
Thus, the correct response is: \[ \text{(−∞,∞)} \]