Question

use the given function f to answer parts (a) through (f) below.
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 below.
c) from the graph, determine the range and any asymptotes of f.
the range of f is .
(type your answer in interval notation.)

Explanation:

Step1: Recall domain of natural - log function

The argument of $\ln(u)$ must be $u>0$. For $f(x)=\ln(x + 3)$, we set $x+3>0$.
$x+3>0$

Step2: Solve the inequality for $x$

Subtract 3 from both sides of the inequality $x+3>0$.
$x>- 3$
In interval notation, the domain is $(-3,\infty)$.

For part (b), since no specific details about the graphs are given in text - based format to analyze further, we skip it for now.

Step3: Recall range of natural - log function

The range of the natural - log function $y = \ln(u)$ is all real numbers. For $y=\ln(x + 3)$, the range is also all real numbers.
In interval notation, the range is $(-\infty,\infty)$.

Step4: Find the vertical asymptote

As $x+3
ightarrow0^{+}$, $x
ightarrow - 3^{+}$. So the vertical asymptote is $x=-3$.

Answer:

a) Domain: $(-3,\infty)$
c) Range: $(-\infty,\infty)$; Vertical asymptote: $x = - 3$