Question

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

Answer 1

Given $f:R\to A$ is defined by

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

Since f is surjection.

So, $R\left(f\right)=\text{Co-domain A}$
Let $y=\frac{x^2}{x^2+1}$

$y=x^2(1-y)$

$\Rightarrow x=\sqrt{\frac{y}{1-y}}$

$\Rightarrow \frac{y}{1-y}\ge 0$

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

So, the range of $f$ is $[0,1)$

So, $A$ is $[0,1)$

Comparing with given A,

$k=0,m=1$

$\Rightarrow k+m=1$