2 functions are defined by f(x)=ax-1 and g(x)=x^2+3x+1, given gf(x)=4x^2+ax+b, find a and b.

Question

Find the value of a and the value of b.

Answer 1

f(x) = ax - 1,  g(x) = x^2 + 3x + 1

gf(x) = 4x^2 + ax + b.

to get gf(x) substitute f(x) in for each value of x in g(x).

gf(x) = (ax - 1)^2 + 3(ax - 1) + 1

gf(x) =  a^2x^2 -2ax + 1 + 3ax - 3 + 1

gf(x) = a^2x^2 +ax - 1  combining common terms

gf(x) = 4x^2 + ax + b    comparing and solving

4 = a^2

a =  +or- 2

b = -1