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

Given functions are f(x) = ax-1

g(x) = x^2+3x+1

gf(x) = g(f(x))

= g(ax-1)

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

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

= (ax)^2+ax-1

= a^2x^2+ax-1

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

4x^2+ax+b = a^2x^2+ax-1

Compare each sides

We can able to write a = 2, b = -1.

The values of a = 2, b = 1.