Question

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

• A. many-one and onto
• B. many-one and into
• C. one-one and onto
• D. one-one and into

Answer 1

Answer: (A)

many-one and onto

$f(-\infty ,-1)⟶(0,e^5)$

$=e^{x^2-3x+2}$

$=e^{x^2-2x-x+2}$

$=e^{x^2-2x-1(x-2)}$

$=e^{x(x-2)-1(x-2)}$

$=e^{(x-1)(x-2)}$

For $two$ values of $x$ $(i.e.$ $x=1,x=2),$ we get $f\left(x\right)=1,$

Hence $f\left(x\right)$ is many one.

Also, $Range\ge codomain.$