Question

Let $f:\{x,\frac{x^2}{1+x^2}:x\in R\}$ be a function from $R\to R$, the range of $f$ is

• A. $[0,1]$
• B. $[0,1)$
• C. $(0,1]$
• D. $(-\infty ,\infty )$

Answer 1

The correct option is B $[0,1)$

Range is the value of $y$

$\therefore y=\frac{x^2}{x^2+1}\ge 0\forall x\in R$

$\Rightarrow (y-1)x^2+y=0$

which is quadratic in $x$ $\therefore$ $y\ne 1$

for $x\in R$

$D\ge 0$

$y(y-1)\le 0,0\le y\le 1$

$\Rightarrow y\in [0,1)$ as $y\ne 1$