Question

If `f(x) = x/6 - 2` and `g(x) = 6x + 12`, how can I show `(f@g)(x) = (g@f)(x)`?

Answer 1

`"see explanation"`

Explanation:

`"if "(f@g)(x)=(g@f)(x)=x" then"`

`f(x)" and "g(x)" are inverse functions of each other"`

`"if we show that "f(x)" and "g(x)" are inverse of each "`
`"other then the above statement should be true"`

`"let "y=x/6-2" and "y=6x+12`

`"rearrange both making x the subject"`

`6y=x-12rArrx=6y+12`

`f^-1(x)=6x+12=g(x)`

`y=6x+12rArrx=1/6(y-12)=y/6-2`

`g^-1x=x/6-2=f(x)`

`rArr(f@g)(x)=(g@f)(x)`

`color(blue)"As a check"`

`(f@g)(x)=f(6x+12)`

`color(white)(xxxxxxx)=(6x+12)/6-2=x+2-2=x`

`(g@f)(x)=g(x/6-2)`

`color(white)(xxxxxxx)=6(x/6-2)+12=x-12+12=x`

`rArr(f@g)(x)=(g@f)(x)`