Question

If `f(x+1) =2x` and `g(3x) = x+6`, how do you find the value of #f^-1(g(f(13)))3?

Answer 1

`f^(-1)(g(f(13)))= 16`

`f^(-1)(g(f(13)))3 = 48` (if the final `3` was intended to be a multiplication factor)

The question is not well formatted.

Explanation:

If `f(x+1) = 2x`
then substituting `a` for `x+1` (which `rarr x=a-1`)
`f(a) = 2(a-1) =2a-2`

Similarly, if `g(3x) = x+6`
then substituting `b` for `3x` (which `rarr x=b/3`)
`g(b)=b/3+6`

By definition of inverse of a function
`f^(-1)(f(a))=a`

`f^(-1)(2a-2)=a`

`f^(-1)(c) =(c+2)/2`

So now we have:
`f^(-1)(g(f(13)))`

substituting `g(f(13))` for `c` above
`color(white)("XXX")=(g(f(13))+2)/2`

substituting `f(13)` for `b` in the equation for `g(b)`
`color(white)("XXX")=((f(13)+6)+2)/2=(f(13)+8)/2`

substituting `13` for `a` in the equation for `f(a)`
`color(white)("XXX")=(2(13)-2+8)/2=32/2=16`