Question

If f(x) = `{{:(x^3 + 1",", x < 0),(x^2 + 1",", x ≥ 0):}`, g(x) = `{{:((x - 1)^(1//3)",", x < 1),((x - 1)^(1//2)",", x ≥ 1):}`, then (gof) (x) is equal to ______.

पर्याय

• x, ∀ x ∈ R

• x – 1, ∀ x ∈ R

• x + 1, ∀ x ∈ R

• None of these

Answer 1

If f(x) = `{{:(x^3 + 1",", x < 0),(x^2 + 1",", x ≥ 0):}`, g(x) = `{{:((x - 1)^(1//3)",", x < 1),((x - 1)^(1//2)",", x ≥ 1):}`, then (gof) (x) is equal to x, ∀ x ∈ R.

Explanation:

Let x < 0.

∴ (gof) (x) = g (f(x)) = g(x³ + 1) = [(x³ + 1) – 1]^1/3  ...(∵ x < 0 `\implies` x³ + 1 < 1)

= (x³)^1/3 = x

Let x ≥ 0.

∴ (gof) (x) = g (f(x)) = g(x² + 1) = ((x² + 1) – 1)^1/2  ...(∵ x ≥ 0 `\implies` x² + 1 ≥ 1)

= (x²)^1/2 = |x| = x ...(∵ x ≥ 0)

∴  (gof) (x) = x ∀ x ∈ R