Problem

For \(f(x)~=~2^{x~-~1}~+~5\) and \(f^{-1}(x)~=~\log_{2}(x~-~5)~+~1\), reflect the asymptote and point \((1,~6)\), then state how the original range becomes the inverse domain.

Neutral card showing f(x)=2^(x−1)+5 and f⁻¹(x)=log₂(x−5)+1, without reflected points, asymptotes, domains, or ranges.

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Features: horizontal asymptote y=5 reflects to inverse vertical asymptote x=5, and point (1,6) reflects to (6,1). Sets: original range (5,∞) becomes inverse domain (5,∞).

Features: horizontal asymptote y=5 remains inverse horizontal asymptote y=5, while point (1,6) reflects to (6,1). Sets: original range (5,∞) remains an output range instead of becoming inverse domain.

Features: horizontal asymptote y=5 reflects to inverse vertical asymptote x=5, and point (1,6) reflects to (6,1). Sets: original range (5,∞) is copied as inverse range rather than inverse domain.

Features: horizontal asymptote y=5 reflects to inverse vertical asymptote x=5, while point (1,6) remains (1,6). Sets: original range (5,∞) correctly becomes inverse domain (5,∞).

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For \(f(x)~=~2^{x~-~1}~+~5\) and \(f^{-1}(x)~=~\log_{2}(x~-~5)~+~1\), reflect the asymptote and point \((1,~6)\), then state how the original range becomes the inverse domain.

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