The Domain and range of the function $f\left(x\right)=\left|x\right|+\left|x+1\right|$respectively

The given function is $f\left(x\right)=\left|x\right|+\left|x+1\right|$

Since it is sum of two modulus funciton, and it has no restrictions, the domain is set of all real numbers

Therefore, the domain is $\boxed{R}$

The given function can be split as below

$f\left(x\right)=\left\{\begin{array}{l}-2x-1 x<-1\\ 1 -1<x<0\\ 2x+1 x>1\end{array}\right.$

The graph of the function $f\left(x\right)=\left|x\right|+\left|x+1\right|$ is as below. 

Observing the function, the minimum value of the funciton is $1$

So that the range of the function is $\boxed{\left[1,\infty \right)}$