Question

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x² – 2x, and g(f(x)) = 4x² + 6x + 1, then the value of f(2) + g(2) is ______.

Options

• 17

• 18

• 19

• 20

Answer 1

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x² – 2x, and g(f(x)) = 4x² + 6x + 1, then the value of f(2) + g(2) is 18.

Explanation:

f(g(x)) = 8x² – 2x, and g(f(x)) = 4x² + 6x + 1

Let f(x) = px² + qx + r

And g(x) = ux + v

Now, f(g(x)) = p(ux + v)² + q(ux + v) + r

= pu²x² + (2puv + qu)x + pv² + qv + r

⇒ pu² = 8, 2puv + qu = –2 and pv² + qv + r = 0

Now, g(f(x)) = u(px² + qx + r) + v

= pux² + qux + v + ur

⇒ pu = 4, qu = 6, v + ur = 1

∴ u = 2, q = 3, p = 2, v = –1, r = 1

∴ f(x) = 2x² + 3x + 1

and g(x) = 2x – 1

Now, f(2) = 2(2)² + 3(2) + 1 = 15

and g(2) = 2(2) – 1 = 3

∴ f(2) + g(2) = 18