Question

How do you find the compositions given `f(x)=8x` and `g(x)=x/8`?

Answer 1

`fog : RR rarr RR,` is defined by, `fog(x)=x.`

`gof : RR rarr RR,` is defined by `gof(x)=x.`

Explanation:

We can easily see that the Domain & Range of funs. `f & g` are `RR.`

Thus, `f, g :RR rarr RR.` We denote by `R_f` & `D_f` the Range & Domain of fun. `f`, resp.

For `fog` may become defined, we must have, `R_g sub D_f,`
& similarly, for `gof`, `R_f sub D_g.` Clearly, these conds. are satisfied, we find that both `fog` & `gof` are defined, and,

`fog : RR rarr RR, gof : RR rarr RR.`

As for formula of `fog`, we have,

`fog(x)=f(g(x))=f(u),` say, where `u=g(x)`
`=8u`.......[since,`f(x)=8x]`
`=8g(x)`.....[since `u=g(x)]`
`=8(x/8)`..........[since `g(x)=x/8`=x#

Thus, `fog : RR rarr RR,` is defined by, `fog(x)=x.`

Similarly, we can show that,

`gof : RR rarr RR,` is defined by `gof(x)=x.`