math problem help

Question

Given that f(x)=2x^2−2x and g(x)=−3x+8 determine each of the following. Make sure to fully simplify your answer.

(a) (f∘g)(x)=   

(b) (g∘f)(x)=   

(c) (f∘f)(x)=   

(d) (g∘g)(x)= 

Answer 1

f(x) = 2x^2-2x, g(x)=-3x +8

a) f(g(x)) = f(-3x+8) = 2(-3x+8)^2 -2(-3x+8) = 2(9x^2-48x+64) +6x-16 =18x^2-96x+128 +6x-16

= 18x^2 -90x +112

b) g(f(x)) = g(2x^2-2x) = -3(2x^2-2x)+8 =-6x^2+6x+8

c) f(f(x)) = f(2x^2-2x) = 2(2x^2-2x)^2 -2(2x^2-2x) = 2(4x^4 -8x^3+4x^2) -4x^2 +4x

= 8x^4 -16x^3 +8x^2 -4x^2 +4x

= 8x^4 -16x^3+4x^2+4x

d) g(g(x)) = g(-3x+8) = -3(-3x+8) + 8 =9x-24+8 = 9x-16

the calculations get a little tedious, easy to make mistakes. One way to check the answer is try an easy integer for x, maybe x=0, then find f(0) =0 and f(f(0)) = f(0)=0, compare that with substituting 0 for x in f(f(x)) = 8x^4-16x^3+4x^2+4x and you get zero again.

try that with a) and it works. Not guaranteed correct, but much more likely

d) g(0)=8 g(8) =-24+8=-16. It works for d)

b) works too for x=0