Let f:{ x, \(\frac{x^2}{1+x^2}\) : x ∈ R} be function from R →

1.	Let f:{ x, \(\frac{x^2}{1+x^2}\) : x ∈ R} be function from R → R, the range of f is(a) [0, 1] (b) [0, 1) (c) (0, 1] (d) (– ∞, ∞)
Answer» Answer : (b) [0, 1) . Let y = \(\frac{x^2}{1+x^2}\) ≥ 0 for all x ∈ R ⇒ y (1 + x)² ≥ x² ⇒ y + yx² – x² ≥ 0 ⇒ (y – 1)x² + y ≥ 0 Let y = \(\frac{x^2}{1+x^2}\) and y ≥ 0 for all x ∈ R. ⇒ \(\frac{x^2}{1+x^2}\) ≥ 0 Also x² < x²+ 1 ⇒ \(\frac{x^2}{1+x^2}\) < 1 ∴ Required range = [0, 1).

Let f:{ x, \(\frac{x^2}{1+x^2}\) : x ∈ R} be function from R → R, the range of f is(a) [0, 1] (b) [0, 1) (c) (0, 1] (d) (– ∞, ∞)

Answer»

Answer : (b) [0, 1)

. Let y = \(\frac{x^2}{1+x^2}\) ≥ 0 for all x ∈ R

⇒ y (1 + x)² ≥ x²

⇒ y + yx² – x² ≥ 0

⇒ (y – 1)x² + y ≥ 0

Let y = \(\frac{x^2}{1+x^2}\) and y ≥ 0 for all x ∈ R.

⇒ \(\frac{x^2}{1+x^2}\) ≥ 0

Also x² < x²+ 1

⇒ \(\frac{x^2}{1+x^2}\) < 1

∴ Required range = [0, 1).