Let f: R arrow R be defined by f(x)= x 1+x^ 2 , x ∈ R . Then the...

f(x)=\frac{x}{1+x^{2}}, x \in R$Let,$y=\frac{x}{1+x^{2}} \Rightarrow y x^{2}-x+y=0 \Rightarrow x=\frac{1 \pm \sqrt{1-4 y^{2}}}{2} \Rightarrow 1-4 y^{2} \geq 0 \Rightarrow 1 \geq 4 y^{2} \Rightarrow|y| \leq \frac{1}{2} \Rightarrow \quad-\frac{1}{2} \leq y \leq \frac{1}{2} \Rightarrow$The range of$f$is \left[-\frac{1}{2}, \frac{1}{2}\right]$.