Answer
$(f\circ g)(x)=6x-3x^{2}-1$
$(g\circ f)(x)=-x^{2}+6x-2$
$(f\circ f)(x)=9x-4$
$(g\circ g)(x)=x^{4}+4x^{3}-6x^{2}+4x$
The domain of $(f\circ g)(x)$, $(g\circ f)(x)$, $(f\circ f)(x)$ and $(g\circ g)(x)$ are $ R $
Work Step by Step
We are given $f(x)=3x-1$ and $g(x)=2x-x^{2}$
Thus we have
$(f\circ g)(x)=f(2x-x^{2})=3(2x-x^{2})-1=6x-3x^{2}-1$
$(g\circ f)(x)=g(3x-1)=2(3x-1)-x^{2}=-x^{2}+6x-2$
$(f\circ f)(x)=f(3x-1)=3(3x-1)-1=9x-4$
$(g\circ g)(x)=g(2x-x^{2})=2(2x-x^{2})-(2x-x^{2})^{2}=4x-2x^{2}-4x^{2}+4x^{3}-x^{4}=x^{4}+4x^{3}-6x^{2}+4x$
The domain of $(f\circ g)(x)$, $(g\circ f)(x)$, $(f\circ f)(x)$ and $(g\circ g)(x)$ are $ R $
