Question 30626
If f(x)=x^2-x and g(x)=-3+x find the following:
(a) f(g(x))  (b)  g(f(x))  (c) f(f(-1))  (d) g(g(-3))

f(x)=x^2-x ----(1)
g(x)=-3+x  ----(2)

(a) f(g(x)) = f(-3+x) using (2)
= f(x-3)  (using additive commutativity we have (-3+x)= (x-3)  )
=f(t) where t = (x-3)----(*)
= t^2-t  using (1)
= (x-3)^2-(x-3)using(*)
=x^2-6x+9-x+3
=x^2-6x-x+9+3
=x^2-7x+12

b)g(f(x)) = g(x^2-x) using (1)
=g(u)  where u = x^2-x  ----(**)
= -3+u   using (2)
= -3+ (x^2-x) using(**)
= x^2-x-3

c) f(f(-1))  
= f[(-1)^2-(-1)]  (using x = -1 in (1)which is f(x)=(x^2-x))
=f[1+1]
=f(2)
= [(2)^2-(2)]  (using x = 2 in (1)which is f(x)=(x^2-x))
= 4-2
=2

(d) g(g(-3))
= g[-3+(-3)] using x = -3 in (2) which is g(x) = -3+x )
=g(-3-3)
=g(-6)
=[-3+(-6)]  using x = -6 in (2) which is g(x) = -3+x )
= (-3-6)
= -9

Answer: a)f(g(x)) = x^2-7x+12
b)g(f(x)) = x^2-x-3
c) f(f(-1))=2 
(d) g(g(-3))= -9