Problem

For \(f(x)~=~\log_{3}(x~+~4)~-~2\) and \(f^{-1}(x)~=~3^{x~+~2}~-~4\), reflect the asymptote and point \((-3,~-2)\), then state how the original domain becomes the inverse range.

Neutral card showing f(x)=log₃(x+4)−2 and f⁻¹(x)=3^(x+2)−4, without reflected points, asymptotes, domains, or ranges.

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Features: original vertical asymptote x=−4 reflects to inverse horizontal asymptote y=−4, and point (−3,−2) reflects to (−2,−3). Sets: original domain (−4,∞) becomes inverse range (−4,∞).

Features: original vertical asymptote x=−4 reflects to inverse horizontal asymptote y=−4, and point (−3,−2) reflects to (−2,−3). Sets: original domain (−4,∞) is copied as inverse domain instead of inverse range.

Features: original vertical asymptote x=−4 remains inverse vertical asymptote x=−4, while point (−3,−2) reflects to (−2,−3). Sets: original domain (−4,∞) remains an input domain rather than inverse range.

Features: original vertical asymptote x=−4 reflects to inverse horizontal asymptote y=−4, while point (−3,−2) remains (−3,−2). Sets: original domain (−4,∞) correctly becomes inverse range (−4,∞).

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For \(f(x)~=~\log_{3}(x~+~4)~-~2\) and \(f^{-1}(x)~=~3^{x~+~2}~-~4\), reflect the asymptote and point \((-3,~-2)\), then state how the original domain becomes the inverse range.

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