Two functions are defined as under : `f(x)={(x+1, x le 1), (2x+1, 1 < x le 2):}` and `g(x)={(x^2, -1 le x le 2), (x+2, 2 le x le 3):}` Find `fog` and `gof`

Explanation

We have `f(g(x))={(g(x)+1",", g(x) le0),(2g(x)+1",",1 lt g(x) le2):}`
Here, g(x) is argument of the function `f(g(x))`
So, we draw the graph of `y=g(x)`.

From the graph it is clear that `g(x) le 1, AA x in [-1,1]`
and `1 lt g(x) le 2, AA x in(1,sqrt(2)]`
` implies f(g(x))={(x^(2)+1",", -1 le x le 1),(2x^(2)+1",",1lt x le sqrt(2)):}`