If `f:R to R` be defined by `f(x) =2x+sinx ` for `x in R`, then check the nature of the function.

Explanation

We have,
`f(x)=2x+sinx, x in R`
`implies f'(x)=2+cosx `
but `-1 le cosx le 1`
`implies 1 le 2 +cosx le 3`
` :. f'(x) gt 0,AA x in R`
Since, `f(x)=2x +sinx` continuously exists, `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.
So, Range of `f(x)` is `R`. Hence, `f(x)` is onto.
Thus f(x) is one-one and onto, i.e., bijective.