Consider a function such that $f\left(x+y\right)=cf\left(x\right)f\left(y\right)$ for all real values of $x$ and $y$.
$f$ always outputs positive values, $f\left(0\right)=\frac{1}{2}$ and $f\left(1\right)=2$. Find:

State the domain for the following function. $$f\left(x\right)=\frac{1}{1-\frac{1}{1+\frac{1}{1-\frac{1}{1-\frac{1}{x+1}}}}}$$ Domain: $\bigl\{x|x\ne$ Check $,x\in \mathbb{R}\bigl\}$

For $\displaystyle f\left(x\right)=\frac{3x+a}{x+b}, f^{-1}\left(1\right)=3$ and $f^{-1}\left(-7\right)=-1$. Find the values of $a$ and $b$.

$a=$ Check $, b=$ Check

Let $\displaystyle f\left(x\right)=\frac{ax+b}{cx-d}\; \left(d\ne 0\right)$ and $\displaystyle g\left(x\right)=\frac{-2x+3}{x-1}$. If $f\left(g\left(x\right)\right)=x$, find $f\left(x\right)$.

$f\left(x\right)=$ Check

For $f\left(x\right)=2ax-5a^2$, find the value(s) of $a$ such that $f^{-1}\left(x\right)=f\left(x\right)$.

$a=$ Check

For $\displaystyle g\left(x\right)=\sqrt{cx+d}, g^{-1}\left(x\right)=\frac{1}{6}x^2-\frac{1}{2}\; \left(x\ge 0\right)$. Find the values of $c$ and $d$.

$c=$ Check $, d=$ Check