The function f: [-1,1]→ R is given by f(x)=displaystylex/(x+2)

The correct answer is Option (3) → one-one only

One-one: let \(f(x_1) = f(x_2)\):

    \( \displaystyle \frac{x_1}{x_1+2} = \frac{x_2}{x_2+2} \)

    \(x_1x_2 + 2x_1 = x_1x_2 + 2x_2\)

    \(x_1 = x_2\)

So f is one-one.

Onto: the codomain is R, so every real number must be an image. Since

    \( \displaystyle f'(x) = \frac{(x+2) - x}{(x+2)^2} = \frac{2}{(x+2)^2} \gt 0 \)

f is increasing on [−1, 1], and its range is

    \( \displaystyle [f(-1), f(1)] = \left[-1, \frac{1}{3}\right] \)

For example, 1 is not an image. So f is not onto, and f is one-one only.