Question

Given f(x)=x^2-4 and g(x)=1-x and k(g(x))=f(x) find k(x) and then k'(x).

Answer 1

k(g(x)) = f(x)

k((1-x)) = x^2 - 4

let m= 1-x
or x = 1-m
then k(m) = (1-m)^2 - 4

k(m) = 1 - 2m + m^2 - 4 = m^2 - 2m - 3
or in terms of x
k(x) = x^2 - 2m - 3
k'(x) = 2x - 2