Download App

RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS1 Mark
MCQ
Let f : R → 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 → 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 "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

If $|A|=|k A|$, where $A$ is a square matrix of order 2 , then sum of all possible values of $k$ is
→
$\int_0^{\frac{\pi}{4}} \frac{e^{\tan x}}{\cos ^2 x} d x=?$
→
Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj} A|=64$. Then $|A|$ is equal to:
→
The value of the definite integral $\int\limits_0^{\pi /2} {\sqrt {\tan \,x\,} } dx$, is
→
If function $f(x)=\left\{\begin{array}{cl}\frac{e^{3 x}-e^{-5 x}}{x} & , \text { if } x \neq 0 \\ k & , \text { if } x=0\end{array}\right.$ is continuous then value of $k$ :
→
Let $f: R \rightarrow R$ be defined as

$f(x)=\left\{\begin{array}{cc}2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x<-1 \\ \left|a x^{2}+x+b\right|, & \text { if }-1 \leq x \leq 1 \\ \sin (\pi x), & \text { if } x>1\end{array}\right.$

If $f(x)$ is continuous on $R,$ then $a+b$ equals ..... .

→
Let $[t]$ denote the greatest integer less than or equal to $t$. Let $\mathrm{f}:[0, \infty) \rightarrow \mathrm{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $\mathrm{f}$ is not continuous. Then $\sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to............
→
If $X$  is a square matrix of order  $ 3 × 3$  and $\lambda $ is a scalar, then adj ($\lambda X)$ is equal to
→
$\int\limits_0^{\frac{\pi }{2}} {\,\,\sqrt {\sin \,2\theta } } \sin\, \theta \,d\theta$ is equal to :
→
Let $f(x)=\min \{[x-1],[x-2], \ldots,[x-10]\}$ where $[ t$ ] denotes the greatest integer $\leq t$.Then$\int_{0}^{10} f(x) d x+\int_{0}^{10}(f(x))^{2} d x+\int_{0}^{10}|f(x)| d x$ is equal to
→