Composite Functions

Let $f:R\to R$ and $g:R\to R$ be defined as $f\left(x\right)=\left\{\begin{array}{cl}x+a,& x<0\\ |x-1|,& x\ge 0\end{array}\right.$ and $g\left(x\right)=\left\{\begin{array}{cc}x+1,& x<0\\ (x-1{)}^2+b,& x\ge 0\end{array}\right.$, where $a, b$ are non-negative real numbers. If $gof\left(x\right)$ is continuous for all $x\in R,$ then $a+b$ is equal to ______ .

Let $f\left(x\right)=\frac{x}{1+x}$ and let $g\left(x\right)=\frac{rx}{1-x}$. Then the number of value(s) of $r$ is such that $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)$ for infinitely many real numbers $x$ is

If $f\left(x\right)=x^2+x+\frac{3}{4}$ and $g\left(x\right)=x^2+ax+1$ be two real functions, then the range of $a$ for which $g\left(f\left(x\right)\right)=0$ has no real solution, is