Let \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) be two real polynomials of degree 2 and 1 respectively. If \(\mathrm{f}(\mathrm{g}(\mathrm{x}))=8 \mathrm{x}^{2}-2 \mathrm{x}\), and \(g(f(x))=4 x^{2}+6 x+1\), then the value of \(f(2)+g(2)\) is ______.

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Sets Relations and Functions

Let

f (x)

and

g (x)

be two real polynomials of degree 2 and 1 respectively. If

f (g (x)) = 8 x^{2} - 2 x

, and

g (f (x)) = 4 x^{2} + 6 x + 1

, then the value of

f (2) + g (2)

is ______.

Answer: 18

Let

g (x) = a x + b

(degree 1) and

f (x) = c x^{2} + d x + e

(degree 2).

From

f (g (x)) = 8 x^{2} - 2 x

f (a x + b) = c (a x + b)^{2} + d (a x + b) + e = 8 x^{2} - 2 x

Comparing coefficients of

x^{2}

c a^{2} = 8

Comparing coefficients of

x

2 c ab + d a = - 2

Comparing constants:

c b^{2} + d b + e = 0

From

g (f (x)) = 4 x^{2} + 6 x + 1

g (c x^{2} + d x + e) = a (c x^{2} + d x + e) + b = 4 x^{2} + 6 x + 1

Comparing:

a c = 4

a d = 6

a e + b = 1

From

c a^{2} = 8

and

a c = 4

a = 2

c = 2

From

a d = 6

d = 3

Testing

b = - 1

a e + b = 2 e - 1 = 1

gives

e = 1

g (x) = 2 x - 1

and

f (x) = 2 x^{2} + 3 x + 1

g (2) = 2 (2) - 1 = 3

f (2) = 2 (4) + 3 (2) + 1 = 8 + 6 + 1 = 15

f (2) + g (2) = 15 + 3 = 18

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Integer answer