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 $f = (0, 1)$
$\Rightarrow A = (0, 1)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Relations and Functions→Relations and Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

In solving the LPP: “minimize f = 6x + 10y subect to constraints $\text{x}\geq6,\text{y}\geq2,2\text{x}+\text{y}\geq10,\text{x}\geq0,\text{y}\geq0$” redundant constraints are:

$\int\limits^\pi_0\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{ dx}=$

Let $f$ and $g$ are twice differentiable functions such that $f(x).g(x) = 1\,\, \forall x \in R$ and $f'$ and $g'$ are never zero, then $\frac{{f^{''}(x)}}{{f(x)}} + \frac{{g^{''}(x)}}{{g(x)}}$ equals-

Let $A$ be the region enclosed by the parabola $y^2=2 x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is ..................

If $\text{f(x)}=\frac{1}{1-\text{x}},$ then the set of points discontinuity of the function $f(f(f(x)))$ is :

$\int_0^1 {\log \sin \left( {\frac{\pi }{2}x} \right)} \,dx = $

If $a \ne p,b \ne q,c \ne r$ and $\left| {\,\begin{array}{*{20}{c}}p&b&c\\{p + a}&{q + b}&{2c}\\a&b&r\end{array}\,} \right|$ =$ 0$, then $\frac{p}{{p - a}} + \frac{q}{{q - b}} + \frac{r}{{r - c}} = $

The area bounded by the curve $y=x^4-2 x^3+x^2+3$ with $x-$ axis and ordinates corresponding to the minima of $y$ is :

$\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}\frac{\text{dx}}{1+\cos2\text{x}}\text{dx}$ is equal to:

The direction ratios of the line perprndicular to the lines $\frac{\text{x}-7}{2}=\frac{\text{y}+17}{-3}=\frac{\text{z}-6}{1}$ and, $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{2}=\frac{\text{z}-4}{-2}$ are proportional to: