Download App

Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions1 Mark
MCQ
Let $f : R \rightarrow R$ be a function defined by $\text{f(x)}=\frac{\text{e}^{|\text{x}|}-\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}.$ Then,
  • A
    $f$ is a bijection.
  • B
    $f$ is an injection only.
  • C
    $f$ is surjection on only.
  • ✓
    $f$ is neither an injection nor a surjection.
✓

Answer

Correct option: D.
$f$ is neither an injection nor a surjection.
$f : R \rightarrow R$
$\text{f(x)}=\frac{\text{e}^{|\text{x}|}-\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$
For $x = -2$ and $-3 \in\text{R}$
$\text{f(-2)}=\frac{\text{e}^{|-2|}-\text{e}^2}{\text{e}^{-2}+\text{e}^2}$
$=\frac{\text{e}^2-\text{e}^2}{\text{e}^{-2}+\text{e}^2}$
$=0$
Hence, for different values of $x$ we are getting same values of $f(x)$
That means, the given function is many one.
Therefore, this function is not injective.
For $x < 0$
$f(x) = 0$
For $x > 0$
$\text{f(x)}=\frac{\text{e}^{\text{x}}-\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$
$=\frac{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{}e^{-\text{x}}}-\frac{2\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$
$=1-\frac{2\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$
The value of $\frac{2\text{e}^{-\text{x}}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$ is always positive.
Therefore, the value of $f(x)$ is always less than $1.$
Numbers more than $1$ are not included in the range but they are included in co$-$domain.
As the codomain is $R.$
$\therefore\ \text{Co-domain}\neq\text{Range}$
Hence, the given function is not onto.
Therefore, this function is not surjective.

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 "MCQ" from Functions→Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The area of the region included between the line $x+y=1$ and the circle $x^2+y^2=1$ is
→
$\frac{\text{d}}{\text{dx}}\Big\{\tan^{-1}\Big(\frac{\cos\text{x}}{1+\sin\text{x}}\Big)\Big\}$ equals:
→
Vectors $(p, q)$ and $(5,1)$ are parallel, if
→
Which of the following is not correct?
→
If p, q are true and r is false statement then which of the following is true statement?
→
If in $\triangle A B C, 2 b^2=a^2+c^2$, then $\frac{\sin 3 B}{\sin B}=$
→
The vector $\vec{\text{b}}=3\hat{\text{i}}+4\hat{\text{k}}$ is to be written as the sum of a vector $\vec{\alpha}$ parallel to $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}$ and a vector $\vec{\beta}$ perpendicular to $\vec{\text{a}}.$ Then $\vec{\alpha}=$
→
If $A^2=-\frac{1}{2}\left[\begin{array}{cc}1 & -4 \\ -1 & 2\end{array}\right]$ then $A=$
→
The sum of the direction cosines of a line which makes equal angles with the positive direction of co-ordinate axes is
→
The angle of intersection of the curves $xy = a^2$ and $x^2 - y^2 = 2a^2$ is:
→