If f(x)=log2(x+2)

Question

, determine which of the following is the domain of f−1(x)
.(1 point)
Responses

(2,∞)
left parenthesis 2 comma infinity right parenthesis

(−∞,∞)
left parenthesis negative infinity comma infinity right parenthesis

(0,∞)
left parenthesis 0 comma infinity right parenthesis

(−2,∞)

Answer 1

To find the domain of the inverse function \( f^{-1}(x) \), we first need to determine the range of the original function \( f(x) = \log_2(x + 2) \).

1. Determine the domain of \( f(x) \):

   • The logarithm function is defined for positive arguments.
   • Thus, \( x + 2 > 0 \) leads to \( x > -2 \).
   • The domain of \( f(x) \) is \( (-2, \infty) \).

2. Find the range of \( f(x) \):

   • As \( x \) approaches \(-2\) from the right (i.e., \( x \to -2^+ \)), \( f(x) = \log_2(x + 2) \) approaches \(-\infty\).
   • As \( x \) approaches \(\infty\), \( f(x) \) approaches \(\infty\).
   • Therefore, the range of \( f(x) \) is \( (-\infty, \infty) \).

3. Determine the domain of the inverse function \( f^{-1}(x) \):

   • The domain of \( f^{-1}(x) \) is equal to the range of \( f(x) \).
   • Since the range of \( f(x) \) is \( (-\infty, \infty) \), this means the domain of \( f^{-1}(x) \) is also \( (-\infty, \infty) \).

Based on this analysis, the correct response is:

\[ \text{(−∞,∞)} \quad \text{(negative infinity, infinity)} \]