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Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions1 Mark
MCQ
$f : R \rightarrow R$ is defined by $\text{f(x)}=\frac{\text{e}^{\text{x}^2}-\text{e}^{-\text{x}^2}}{\text{e}^{\text{x}^2}+\text{e}^{-\text{x}^2}}$ is:
  • A
    One$-$one but not onto.
  • B
    Many$-$one but onto.
  • C
    One$-$one and onto.
  • ✓
    Neither one$-$one nor onto.
✓

Answer

Correct option: D.
Neither one$-$one nor onto.
We have,
$\text{f(x)}=\frac{\text{e}^{\text{x}^2}-\text{e}^{-\text{x}^2}}{\text{e}^{\text{x}^2}+\text{e}^{-\text{x}^2}}$
Here, $-2,2\in\text{R}$
Now, $2\neq-2$
But, $f(2) = f(-2)$
Therefore, function is not one-one.
And,
The minimum value of the function is $0$ and maximum value is $1.$
That is range of the function is $[0, 1]$ but the co$-$domain of the function is given $R.$
Therefore, function is not onto.
$\therefore$ function is neither one$-$one nor onto.

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