Which of the following is the range of f−1(x)

Question

𝑓
−
1
(
𝑥
)
if f(x)=log(x)+3
𝑓
(
𝑥
)
=
log
⁡
(
𝑥
)
+
3
?(1 point)
Responses

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

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

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

(3,∞)

Answer 1

To find the range of the inverse function \( f^{-1}(x) \) given \( f(x) = \log(x) + 3 \), we first determine the range of \( f(x) \).

1. Function Analysis: The function \( f(x) = \log(x) + 3 \) is defined for \( x > 0 \) (since the logarithm function is only defined for positive values).

2. Behavior of \( \log(x) \): The logarithm function \( \log(x) \) approaches negative infinity as \( x \) approaches 0 and increases without bound as \( x \) increases. Therefore:

   • As \( x \to 0^+ \), \( \log(x) \to -\infty \).
   • As \( x \to +\infty \), \( \log(x) \to +\infty \).

3. Finding the Range of \( f(x) \): Adding 3 to \( \log(x) \) shifts the entire graph up by 3. Thus:

   • As \( x \to 0^+ \), \( \log(x) + 3 \to -\infty + 3 = -\infty \).
   • As \( x \to +\infty \), \( \log(x) + 3 \to +\infty + 3 = +\infty \).

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

4. Range of the Inverse Function: The range of the inverse function \( f^{-1}(x) \) corresponds to the domain of \( f(x) \). Since \( f(x) \) is defined for \( x > 0 \), the domain of \( f(x) \) is \( (0, +\infty) \).

Therefore, the range of \( f^{-1}(x) \) is: \[ (0, +\infty) \]

Based on the options given, the correct response is: (0,∞).