Function f and g are defined by

Question

f:x→4-x and g:x→px^2+q where p and q are constants.Given that the composite function gf:x→x^2-8x+11,find
the values of p and q.

Answer 1

f(x) = 4-x
g(x) = px^2 + q

g(f(x))
= g(4-x)
= p(4-x)^2 + q
= p(16 - 8x + x^2) + q
= px^2 - 8p x + 16p+q

if px^2 - 8p x + 16p+q = x^2 - 8x + 11

then p = 1
16p+q = 11
16 + q = 11
q = -5

and -8p = -8
p = 1, which checks out