Given the function
[tex]f(x)=2^{3-x}-2[/tex]
Substitute y = f(x) into the function above
[tex]y=2^{3-x}-2[/tex]
Make 'x' the subject of the formula
[tex]y+2=2^{3-x}[/tex]
Take the log₂ of both sides
[tex]\begin{gathered} \log _2(y+2)=\log _22^{(3-x)} \\ \log _2(y+2)=3-x\log _22 \\ \end{gathered}[/tex]
Note:
[tex]\log _22=1[/tex]
Therefore,
[tex]\begin{gathered} \log _2(y+2)=(3-x)\times1 \\ \log _2(y+2)=3-x \end{gathered}[/tex]
Isolating 'x'
[tex]x=3-\log _2(y+2)[/tex]
Replace 'x' with 'y'
[tex]y=3-\log _2(x+2)[/tex]
Hence,
[tex]f^{-1}(x)=3-\log _2(x+2)[/tex]
The correct option is Option 2.
