Question

How do you find (f o g)(x) and its domain, (g o f)(x) and its domain, (f o g)(-2) and (g o f)(-2) of the following problem `f(x) = x^2 – 1`, `g(x) = x + 1`?

Answer 1

Given
`color(white)("XXX")f(color(blue)(x))=color(blue)(x)^2-1`
and
`color(white)("XXX")g(color(red)(x))=color(red)(x)+1`

Note that `(f@g)(x)` can be written `f(g(x))`
and that `(g@f)(x)` can be written `g(f(x))`

`(f@g)(x) = f(color(blue)(g(x))) = color(blue)(g(x))^2-1`
`color(white)("XXXXXX")=(color(blue)(x+1))^2-1`
`color(white)("XXXXXX")=x^2+2x`
Since this is defined for all Real values of `x`,
the Domain of `(f@g)(x)` is all Real values.
(although it wasn't asked for, the Range would be `[-1,+oo)`)

Similarly
`(g@f)(x)=g(color(red)(f(x)))+1`
`color(white)("XXXXXX")=g(color(red)(x^2-1))`
`color(white)("XXXXXX")=color(red)(x^2-1)+1`
`color(white)("XXXXXX")=x^2`
Again, this is defined for all Real values of `x`
so the Domain is all Real values.
(but the Range is `[0,+oo)`)