Question

use the given function f to answer parts (a) through (f).
$f(x)=2+\ln x$

(c) from the graph, determine the range and any asymptotes of f.
the range of f is $(-\infty,\infty)$
(type your answer in interval notation )
determine the vertical asymptote of f, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the vertical asymptote of f is $x = 0$ (simplify your answer.)
b. there is no vertical asymptote

(d) find $f^{-1}$, the inverse of f.
$f^{-1}(x)=\square$ (simplify your answer.)

Explanation:

Step1: Swap x and y in the function

Given \( f(x) = 2 + \ln x \), let \( y = 2 + \ln x \). Swap \( x \) and \( y \) to get \( x = 2 + \ln y \).

Step2: Solve for y

Subtract 2 from both sides: \( x - 2 = \ln y \).
Convert from logarithmic form to exponential form (since \( \ln y \) is base \( e \) logarithm, \( \ln a = b \) implies \( a = e^b \)): \( y = e^{x - 2} \).

Answer:

\( f^{-1}(x)=e^{x - 2} \)