Question

The function $f:(-\infty ,-1]\to (0,e^5]$ defined by $f\left(x\right)=e^{x^3+3x+2}$ is

• A. a one-one function
• B. a many-one function
• C. an into function
• D. an onto function

Answer 1

Answer: (A), (C)

an into function

$f\left(x\right)=e^{x^3+3x+2}$
$f^{\prime }\left(x\right)=e^{x^3+3x+2}\cdot (3x^2+3)>0\forall x\in (-\infty ,-1]$
$\Rightarrow f$ is strictly increasing on $(-\infty ,-1]$
So $f$ is a one-one function.
Now,
$y=e^{x^3+3x+2}$


By graph, for domain $x\in (a,b)$ range of $y=e^x$ is $(e^a,e^b)$
So, range of $f\left(x\right)$ is $\left(f\right(x\to -\infty ),f(-1\left)\right]=(0,e^{-2}]\ne (0,e^5]$
So, $f$ is not an onto function.