You need to evaluate the domain and the range of the resulting functions `fog` and `gof` .

Considering `f(x) = 4x + 1, g(x) = x^2` , yields:

`(fog)(x) = f(g(x)) => (fog)(x) = 4g(x) + 1 => (fog)(x) = 4x^2 + 1`

**Hence, evaluating the domain of definition of the resulting function `(fog)(x) = 4x^2 + 1` is the real set `R` .**

You may evaluate the range of the function such that:

`4x^2 + 1 = y => x^2 = (y - 1)/4 => x_(1,2) = +-sqrt(y-1)/2`

You should notice that the numerator `sqrt(y-1)` has a real value if `y - 1 >= 0 => y >= 1` .

**Hence, evaluating the range of the resulting function yields** `[1,oo)` .

`(gof)(x) = g(f(x)) => (gof)(x) = (f(x))^2 => (gof)(x) = (4x + 1)^2`

**You need to evaluate the domain and the range of the resulting function `(gof)(x) = (4x + 1)^2` , hence, the domain is the real set R and the range is `[0,oo)` .**