Question

If functions g and h are defined as

g(x) = `{{:(x^2 + 1, x∈Q),(px^2, x\cancel(∈)Q):}`

and h(x) = `{{:(px, x∈Q),(2x + q, x\cancel(∈)Q):}`

If (g + h)(x) is continuous at x = 1 and x = 3, then 3p + q is ______.

Options

• 1.00

• 2.00

• 3.00

• 4.00

Answer 1

If functions g and h are defined as

g(x) = `{{:(x^2 + 1, x∈Q),(px^2, x\cancel(∈)Q):}`

and h(x) = `{{:(px, x∈Q),(2x + q, x\cancel(∈)Q):}`

If (g + h)(x) is continuous at x = 1 and x = 3, then 3p + q is 2.00.

Explanation:

g(x) + h(x) = `{{:(x^2 + 1 + px, x∈Q),(px^2 + 2x + q, x\cancel(∈)Q):}`

If (g + h)(x) is continuous at x = 1 and x = 3

⇒ 1 and 3 are roots of equation

⇒ x² + 1 + px = px² + 2x + q

⇒ x²(1 – p) + x(p – 2) + 1 – q = 0

Now `(2 - p)/(1 - p)` = 4 and `(1 - q)/(1 - p)` = 3

⇒ p = `2/3`, q = 0

⇒ 3p + q = 2