Question

Let `f(x)=8x` and `g(x)=x/8`, how do you find each of the compositions and domain and range?

Answer 1

`f(g(x))=xcolor(white)("XXX")`and`color(white)("XXX")g(f(x))=x`
Both functions and their compositions have Domains and Ranges of `(-oo,+oo)`

Explanation:

Sometimes the use of `x` in multiple definitions can cause confusion, so let's re-write the base equations as:
`color(white)("XXX")f(a)=8a`
and
`color(white)("XXX")g(b)=b/8`

So, replacing `a` with `g(b)` we have
`color(white)("XXX")f(g(b)) = 8*g(b) = 8*b/8 = b`
(or, using the original `x` as the variable: `f(g(x))=x`)

Similarly `g(f(x)) = x`

#{ (f(x) = 8x), (g(x) = (x)/(8)), (f(g(x))=x), (g(f(x)))=x :}#
`color(white)("XXX")`are all defined for all values of`x`
and
therefore have Domains of all Real values, `(-oo,+oo)`

Replacing, for example `f(x)` with `y`
we can see that `f(x)=8x <=>y/8=x`
which is defined for all Real values of `y`.
(this process can be done for all of the functions)

So the Ranges of these functions is also all Real values `(-oo,+oo)`