`f(x)={x+1, x<0,x^2, xgeq0` and `g(x)={x^3, x<1, 2x-1, xgeq1` Then find `f(g(x))` and find its domain and range.

Explanation

`f(g(x))={(g(x)+1",", g(x) lt0),({g(x)}^(2)",",g(x) ge0):}`
`f(x)={(x^(3)+1",",x^(3) lt0", " x lt1),(2x-1+1",",2x-1 lt 0", "x ge1),((x^(3))^(2)",",x^(3) ge0", " x lt1),((2x-1)^(2)",",2x-1 ge 0", "x ge1):}`
`={(x^(3)+1",", x lt0),(x^(6)",",0le x lt1),((2x-1)^(2)",", x ge1):}`
Clearly, domain of function is R.
For `x lt 0, x^(3) +1 in (-oo,1).`
For `0le x lt 1, x^(6) in [0,1).`
For ` x ge 1, (2x-1)^(2) in [1, oo).`
Hence, the range is R.