Question

Assertion :The function $f:R\to [0,1)$ defined by $f\left(x\right)=\frac{x^2}{x^2+1}$ is a surjection. Reason: For surjection, Range of $f\left(x\right)=$ codomain of $f\left(x\right)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Defination for onto function, codomain of $f=$ range of $f$

Let $f\left(x\right)=ytheny\ge 0andy=f\left(x\right)=\frac{x^2}{1+x^2}$

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

$\Rightarrow x^2=\frac{y}{1-y}\ge 0duetoL.H.S=x^2$

$\Rightarrow \frac{y-0}{y-1}\le 0(y\ne 0)$

$\Rightarrow 0\le y<1$

$\Rightarrow$ codomain of $f=$range of $f$ , As

$f:R\to [0,1)$