Question

If f(x)=log2(x+2)
, determine which of the following is the domain of f−1(x)
.(1 point)
Responses

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

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

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

(−∞,∞)

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. **Identify the domain of \( f(x) \)**:
The function \( f(x) \) is defined for \( x + 2 > 0 \) (since the logarithm is only defined for positive arguments). This means:
\[
x > -2
\]
So, the domain of \( f(x) \) is \( (-2, \infty) \).

2. **Determine the range of \( f(x) \)**:
As \( x \) approaches \( -2 \) from the right, \( f(x) = \log_2(x+2) \) approaches \( \log_2(0) \), which is \( -\infty \). As \( x \) approaches \( \infty \), \( f(x) \) approaches \( \infty \).

Therefore, the range of \( f(x) \) is:
\[
(-\infty, \infty)
\]

3. **Find the domain of \( f^{-1}(x) \)**:
The domain of the inverse function \( f^{-1}(x) \) is equal to the range of the original function \( f(x) \).

Thus, the domain of \( f^{-1}(x) \) is \( (-\infty, \infty) \).

The correct option is:
**(−∞,∞)**