Answer
a) $x^2+2x-2$
b) $x^2+4$
c) $x^2-5x+7$
d) $x^2+x-2$
e) $1$
f) $4$
g) $x-9$
Work Step by Step
a) $x^2+x+1+(x-3)=$
$x^2+x+1+x-3=$
$x^2+2x-2$
b) $x^2+x+1-(x-3)=$
$x^2+x+1-x+3=$
$x^2+4$
c) $f$ º $g=f(g(x)=$
$(x-3)^2+(x-3)+1=$
$x^2-6x+9+x-3+1=$
$x^2-5x+7$
d) $g$ º $f=g(f(x)=$
$(x^2+x+1)-3=$
$x^2+x-2$
e) The equation of the composition $f(g(x))$ was already found above, so one just need to substitute the $x$ for $2$ to find $f(g(2))$:
$2^2-5(2)+7=$
$4-10+7=$
$1$
f) The equation of the composition $g(f(x))$ was already found above, so one just need to substitute the $x$ for $2$ to find $g(f(2))$:
$2^2+2-2=$
$4$
g) $g$ º $g$ º $g=g(g(g(x)))$ To find this, one first finds $g(g(x))$:
$g(g(x))=(x-3)-3=x-6$ Now, one can find the solution:
$g(g(g(x)))=g(x-6)=(x-6)-3=x-9$
