g(x)=3ex

Question

𝑔
(
𝑥
)
=
3
𝑒
𝑥
. Which of the following is the domain of g−1(x)
𝑔
−
1
(
𝑥
)
?(1 point)
Responses

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

(3,∞)
(
3
,
∞
)
left parenthesis 3 comma infinity right parenthesis

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

(0,∞)

Answer 1

To determine the domain of the inverse function \( g^{-1}(x) \), we first need to analyze the given function \( g(x) = 3 e^x \).

1. Finding the Range of \( g(x) \):

   • The exponential function \( e^x \) is always positive for all real \( x \) and approaches 0 as \( x \to -\infty \), while it approaches \( +\infty \) as \( x \to +\infty \).
   • Therefore, \( g(x) = 3 e^x \) will always be positive and approaches 0 (but never reaches it) as \( x \to -\infty\).
   • Thus, the range of \( g(x) \) is \( (0, \infty) \).

2. Finding the Domain of the Inverse Function \( g^{-1}(x) \):

   • The domain of the inverse function \( g^{-1}(x) \) corresponds to the range of the original function \( g(x) \).
   • Since we established that the range of \( g(x) \) is \( (0, \infty) \), the domain of the inverse function \( g^{-1}(x) \) is also \( (0, \infty) \).

The correct response is:

\[ (0, \infty) \]