Question

If the function $f:R\to$ A given by $f\left(x\right)=\frac{x^2}{x^2+1}$ is a surjection, then A is

• A. $R$
• B. $[0,1]$
• C. $(0,1]$
• D. $[0,1)$

Answer 1

Answer: (D)

$[0,1)$

As 'f' is surjective,

Range of f = co-domain of 'f'

$⟹$ A = range of 'f'

Since, $f\left(x\right)=\frac{x^2}{x^2+1}$

$⟹$ $y=\frac{x^2}{x^2+1}$

$⟹$ $y(x^2+1)=x^2$

$⟹$ $(y-1)x^2+y=0$

$⟹$ $x^2=\frac{-y}{y-1}$

$⟹$ $x=\sqrt{\frac{y}{1-y}}$

$⟹$ $\frac{y}{1-y}\ge 0$

$⟹$ $yϵ[0,1)$

$⟹$ $A=[0,1)$