Question

use the given function f to answer parts (a) through (f)
$f(x)=\ln(x + 3)$
(a) find the domain of f.
the domain of f is $(-3,\infty)$
(type your answer in interval notation.)
(b) graph f. choose the correct graph.
(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 an
a. the vertical asymptote of f is $x = -3$
(simplify your answer.)
b. there is no vertical asymptote.
(d) find $f^{-1}$, the inverse of f.
$f^{-1}(x)=e^{x - 3}$ (simplify your answer.)

Explanation:

Step1: Find the inverse function

Let \( y = \ln(x + 3) \).
To find the inverse, we first swap \( x \) and \( y \): \( x=\ln(y + 3) \).

Step2: Solve for \( y \)

Using the property that if \( x=\ln(a) \), then \( a = e^{x}\) (since \( y=\ln(x)\) and \( x = e^{y}\) are inverse - function relationships).
From \( x=\ln(y + 3) \), we get \( y+3=e^{x}\).
Then \( y=e^{x}-3\).

Answer:

\(f^{-1}(x)=e^{x}-3\)