Consider a function f: [ - 1,1 ] R where f(x) = _1 ^ - 1 x + _3 (… - Vidyadip

MCQ

Consider a function $f:\left[ { - 1,1} \right] \to R$ where $f(x) = {\alpha _1}{\sin ^{ - 1}}x + {\alpha _3}\left( {{{\sin }^{ - 1}}{x^3}} \right) + ..... + {\alpha _{(2n + 1)}}{({\sin ^{ - 1}}x)^{(2n + 1)}} - {\cot ^{ - 1}}x$ Where $\alpha _i\ 's$ are positive constants and $n \in N < 100$ , then $f(x)$ is

A
one-one and onto
✓
one-one and into
C
many one and onto
D
many one and into

✓

Answer

Correct option: B.

one-one and into

b
$ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\alpha_{1}}{\sqrt{1-\mathrm{x}^{2}}}+ \frac{\alpha_{3}\left(3 \sin ^{-1} \mathrm{x}\right)^{2}}{\sqrt{1-\mathrm{x}^{2}}}+\ldots \ldots $

$+\frac{(2 \mathrm{x}+1)\left(\sin ^{-1} \mathrm{x}\right)^{2 n}}{\sqrt{1-\mathrm{x}^{2}}}+\frac{1}{\left(1+\mathrm{x}^{2}\right)} $

$\frac{\mathrm{d} y}{\mathrm{dx}}>0$ and function is one one

but range $\neq$ codomain

$\Rightarrow$ into fumction

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