Question

F(x) = 2^3x-1 - 2find the inverse function of f(x)

Answer 1

Given the function

`f(x)=2^(3-x)-2`

Substitute y = f(x) into the function above

`y=2^(3-x)-2`

Make 'x' the subject of the formula

`y+2=2^(3-x)`

Take the log₂ of both sides

`\begin{gathered} \log _2(y+2)=\log _22^((3-x)) \\ \log _2(y+2)=3-x\log _22 \\ \end{gathered}`

Note:

`\log _22=1`

Therefore,

`\begin{gathered} \log _2(y+2)=(3-x)*1 \\ \log _2(y+2)=3-x \end{gathered}`

Isolating 'x'

`x=3-\log _2(y+2)`

Replace 'x' with 'y'

`y=3-\log _2(x+2)`

Hence,

`f^(-1)(x)=3-\log _2(x+2)`

The correct option is Option 2.