MCQ
The function $f:R \to R$ defined by $f(x) = {e^x}$ is
- AOnto
- BMany-one
- ✓One-one and into
- DMany one and onto
Let ${x_1},\,{x_2} \in R$ and $f({x_1}) = f({x_2})$ or ${e^{{x_1}}} = {e^{{x_2}}}$ or ${x_1} = {x_2}$.
Therefore $f$ is one-one. Let $f(x) = {e^x} = y$.
Taking $log$ on both sides, we get $x = \log y$.
We know that negative real numbers have no pre-image or the function is not onto and zero is not the image of any real number.
Therefore function $f$ is into.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(A)$ There exist $r , s \in R$, where $r < s$, such that $f$ is one-one on the open interval $( r , s )$
$(B)$ There exists $x 0 \in(-4,0)$ such that $\left| f ^{\prime}\left( x _0\right)\right| \leq 1$
$(C)$ $\lim _{x \rightarrow \infty} f(x)=1$
$(D)$ There exists a $\in(-4,4)$ such that $f(a)+f^{\prime \prime}(a)=0$ and $f^{\prime}(a) \neq 0$