function f:x=x-3 and g is such that fg:x=x^2+2x-2. find the function g. hence show that fg(x)not= to gf(x)?

Question

hiii, try to solve this for me...thx

Answer 1

f(x) = x - 3

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

g(x) has to be a quadratic, so
let g(x) = ax^2 + bx + c. Then

f(g(x)) = f(ax^2 + bx + c) = ax^2 + bx + c - 3

Equate this to what we know;

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

Therefore,

ax^2 + bx + {c - 3} = x^2 + 2x - 2

Equate the coefficients.

a = 1
b = 2
c - 3 = -2

a = 1, b = 2, c = -1.

Therefore,

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

g(f(x)) = g(x - 3) = (x - 3)^2 + 2(x - 3) - 1
= x^2 - 6x + 9 + 2x - 6 - 1
= x^2 - 4x - 2, which is not equal to x^2 + 2x - 2.

f(g(x)) is not equal to g(f(x)).

Answer 2

f(x)=x-3
f(g(x))=x^2+2x-2=(x+1)^2-3
if x+1=k then you have
f(k)=k^2-3
that it means g(x)=k^2=(x+1)^2

Second part:
f(g(x))=x^2+2x-2 , f(x)=x-3
g(f(x))=g(x-3)=((x-3)+1)^2=(x-2)^2=x^2-4x+4
so you can see that f(g(x)) is not equal to
g(f(x))