For the given functions f and g , find the indicated composition.

Question

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

(f.g)(-2)

Answer 1

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

g(x) = x^2-2x+2

g(-2) =(-2)^2 -2(-2) +2

= 4+4+2 =10

find f(g(-2)) = f(10)

=10^2 -2(10)+1

=100-20+1 = 81

f(g(-2)) = 81

or in a much more cumbersome method:

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

=x^4 +4x^3 +8x^2-8x +4 -2x^2 +4x +1

= x^4 -4x^3 +6x^2 -4x +1

f(g(-2)) = (-2)^4 -4(-2)^3 +6(-2)^2-4(-2) +1

= 16+32+24+8+5=81

f(g(-2))=81

that's a check on the answer

work a problem 2 different ways and come up with the same answer

that gives a lot of confidence it must be right

Answer 2

(f∘g)(-2) = f(g(-2))

g(-2) = (-2)^2 - 2(-2) + 2

= 4 + 4 + 2

= 10

Since g(-2) = 10, we plug in 10 for f(g(-2))

f(10) = (10)^2 - 2(10) + 1

= 100 - 20 + 1

= 81

Hence (f∘g)(x) = 81