Answer
a) $(f+g)(x)=x^{2}+2x-2$
b) $(f-g)(x)=x^{2}+4$
c) $(f \circ g)(x)=x^{2}-5x+7$
d) $(g \circ f)(x)=x^{2}+x-2$
e) $f (g(2))=1$
f) $g (f(2))=4$
g) $(g\circ g \circ g)=x-9$
Work Step by Step
We are given $f(x)=x^{2}+x+1$ and $g(x)=x-3$
a) $(f+g)(x)=x^{2}+x+1+x-3=x^{2}+2x-2$
b) $(f-g)(x)=x^{2}+x+1-x+3=x^{2}+4$
c) $(f \circ g)(x)=f(x-3)=(x-3)^{2}+x-3+1=x^{2}-6x+9+x-3+1=x^{2}-5x+7$
d) $(g \circ f)(x)=g(x^{2}+x+1)=x^{2}+x+1-3=x^{2}+x-2$
e) $f (g(2))=f(-1)=(-1)^{2}-1+1=1$
f) $g (f(2))=g(7)=7-3=4$
g) $(g\circ g \circ g)=g(g(x-3))=g(x-6)=x-6-3=x-9$
