use the given function f to answer parts (a) through (f)\n$f(x)=\\ln(x + 3)$\n(a) find the domain of f.\nthe…

use the given function f to answer parts (a) through (f)\n$f(x)=\\ln(x + 3)$\n(a) find the domain of f.\nthe domain of f is $(-3,\\infty)$\n(type your answer in interval notation.)\n(b) graph f. choose the correct graph.\n(c) from the graph, determine the range and any asymptotes of f.\nthe range of f is $(-\\infty,\\infty)$\n(type your answer in interval notation.)\ndetermine the vertical asymptote of f, if it exists. select the correct choice an\na. the vertical asymptote of f is $x = -3$\n(simplify your answer.)\nb. there is no vertical asymptote.\n(d) find $f^{-1}$, the inverse of f.\n$f^{-1}(x)=e^{x - 3}$ (simplify your answer.)

use the given function f to answer parts (a) through (f)\n$f(x)=\\ln(x + 3)$\n(a) find the domain of f.\nthe domain of f is $(-3,\\infty)$\n(type your answer in interval notation.)\n(b) graph f. choose the correct graph.\n(c) from the graph, determine the range and any asymptotes of f.\nthe range of f is $(-\\infty,\\infty)$\n(type your answer in interval notation.)\ndetermine the vertical asymptote of f, if it exists. select the correct choice an\na. the vertical asymptote of f is $x = -3$\n(simplify your answer.)\nb. there is no vertical asymptote.\n(d) find $f^{-1}$, the inverse of f.\n$f^{-1}(x)=e^{x - 3}$ (simplify your answer.)

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)