Answer :
[tex]f\left( x \right) =\frac { 1 }{ 2 } \cdot { 3 }^{ x }\\ \\ \frac { 1 }{ 2 } \cdot { 3 }^{ x }=y[/tex]
[tex]\\ \\ 2\cdot \frac { 1 }{ 2 } \cdot { 3 }^{ x }=2\cdot y\\ \\ { 3 }^{ x }=2y\\ \\ \log _{ 3 }{ \left( 2y \right) } =x[/tex]
[tex]\\ \\ \therefore \quad { f }^{ -1 }\left( x \right) =\log _{ 3 }{ \left( 2x \right) }[/tex]
[tex]\\ \\ \therefore \quad { f }^{ -1 }\left( 7 \right) =\log _{ 3 }{ \left( 2\cdot 7 \right) } =\log _{ 3 }{ 14 } [/tex]
[tex]\\ \\ 2\cdot \frac { 1 }{ 2 } \cdot { 3 }^{ x }=2\cdot y\\ \\ { 3 }^{ x }=2y\\ \\ \log _{ 3 }{ \left( 2y \right) } =x[/tex]
[tex]\\ \\ \therefore \quad { f }^{ -1 }\left( x \right) =\log _{ 3 }{ \left( 2x \right) }[/tex]
[tex]\\ \\ \therefore \quad { f }^{ -1 }\left( 7 \right) =\log _{ 3 }{ \left( 2\cdot 7 \right) } =\log _{ 3 }{ 14 } [/tex]
