The function `f:(-oo,-1)vec(0, e^5)` defined by `f(x)=e^x^(3-3x+2)` is many one and onto many one and into one-one and onto one-one and into

Options

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

Answer

D

Explanation

`f(x)=e^(x^(3)-3x+2)`
Let `g(x)=x^(3)-3x+2,`
` g'(x)=3x^(2)-3=3(x^(2)-1)`
` ge 0 " for " x in (-oo,-1]`
Therefore, `f(x)` is increasing function. Hence, `f(x)` is one-one.
Now, the range of `f(x)" is "(0,e^(4)].`
But co-domain is `(0,e^(5)].`
Hence, `f(x)` is an into function.