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If f:R to R be defined by f(x) =2x+sinx ...

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

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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.
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