Question

$f\left(x\right)=\{\begin{array}{cc}x+1& x<0\\ x^2& x\ge 0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}{x}^3& x<1\\ 2x-1& x\ge 1\end{array}$
Then find $f\left(g\right(x\left)\right)$ and find its domain and range.

Answer 1

$f\left(g\left(x\right)\right)=\{\begin{array}{cc}g\left(+1\right)& g\left(x\right)<0\\ 2\left(g\left(x\right)\right)-1& g\left(\ge 0\right)\end{array}$

$=\{\begin{array}{c}{x}^3+1\\ {\left(x^3\right)}^2\\ {\left(2x-1\right)}^2\end{array}\begin{array}{c}x<0\\ 0\le x<1\\ x\ge 1\end{array}$

$-\infty <x<0or0\le x<1or1\cdot \le x$

$Domain=R,Range\equiv R$

$Domain\equiv R\equiv \left(-\infty ,\infty \right)$