MCQ
Function $f:R \to R,\;f(x) = {x^2} + x$ is
- AOne-one onto
- BOne-one into
- CMany-one onto
- ✓Many-one into
hence $f(x)$ is many one.
But there is no pre-image of $ - 1$.
Hence $f(x)$ is into function.
So function is many-one into.
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$(\sqrt 3 - 1)\,\sin \,\theta \, + \,(\sqrt 3 + 1)\,\cos \theta \, = \,2$ is
$\frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1$
Let $S=\left\{y_{\alpha \beta}(x): \alpha, \beta \in R \right\}$. Then which of the following functions belong(s) to the set $S$ ?
$(A)$ $f( x )=\frac{ x ^2}{2} e ^{- x }+\left( e -\frac{1}{2}\right) e ^{- x }$
$(B)$ $f( x )=-\frac{ x ^2}{2} e ^{- x }+\left( e +\frac{1}{2}\right) e ^{- x }$
$(C)$ $f( x )=\frac{ e ^{ x }}{2}\left( x -\frac{1}{2}\right)+\left( e -\frac{ e ^2}{4}\right) e ^{- x }$
$(D)$ $f( x )=\frac{ e ^{ x }}{2}\left(\frac{1}{2}- x \right)+\left( e +\frac{ e ^2}{4}\right) e ^{- x }$