If the function f : R A given by f(x)= x^2 x^2+1 is a surjection,… - Vidyadip

MCQ

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

A
$R$
B
$[0, 1]$
C
$(0, 1)$
✓
$(0, 1)$

✓

Answer

Correct option: D.

$(0, 1)$

As $f$ is surjective, range of $f = co-$domain of $f$
$\Rightarrow A =$ range of $f$
$=\frac{\text{x}^2}{\text{x}^2+1},$
$\text{y}=\frac{\text{x}^2}{\text{x}^2+1}$
$\Rightarrow\ \text{y}(\text{x}^2+1)$
$\Rightarrow\ \text{x}^2=\frac{-\text{y}}{(\text{y}-1)}$
$\Rightarrow\ \text{x}=\sqrt{\frac{\text{y}}{(1-\text{y})}}$
$\Rightarrow\ \frac{\text{y}}{(1-\text{y})}\geq0$
$\Rightarrow\ \text{y}\in[0,1)$
$\Rightarrow $ Range of $f = (0, 1)$
$\Rightarrow A = (0, 1)$

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