If ` f : R to R ` be a function such that `f(x)=x^(3)+x^(2)+3x +sinx,` then discuss the nature of the function.

Explanation

`f(x)=x^(3) +x^(2)+3x+sin x`
` :. f'(x) = 3x^(2)+2x+3 +cosx`
`=3[(x+(1)/(3))^(2)+(8)/(9)]-(-cosx)`
Now `3[(x+(1)/(3))^(2)+(8)/(9)]_("min")=(8)/(3)` and (-cos x) has maximum value '1'.
Thus `f'(x) gt 0`.
Hence, f(x) is one-one.
Also, when x approaches to infinity, f(x) approaches to infinity and when x approaches to negative infinity, f(x) approaches to negative infinity.
Also, f(x) continuously exists for all real x. So, f(x) takes all real values.
Hence, range of the f(x) is R.
Therefore, f(x) is onto.