Let the function f:R to R be defined by f(x) | vedclass

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(A) Given the function $f(x) = 2x + \sin x$ for $x \in R$.To check for one-to-one,we find the derivative: $f'(x) = 2 + \cos x$.Since $-1 \le \cos x \l

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One-to-one and onto

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(A) Given the function $f(x) = 2x + \sin x$ for $x \in R$.To check for one-to-one,we find the derivative: $f'(x) = 2 + \cos x$.Since $-1 \le \cos x \le 1$,we have $f'(x) = 2 + \cos x \ge 2 - 1 = 1 > 0$.Since $f'(x) > 0$ for all $x \in R$,the function $f(x)$ is strictly increasing,which implies it is one-to-one.To check for onto,we observe the range of $f(x)$. Since $f(x)$ is a continuous function and $\lim_{x 	o \infty} f(x) = \infty$ and $\lim_{x 	o -\infty} f(x) = -\infty$,the range of $f(x)$ is $(-\infty, \infty) = R$.Since the range equals the codomain,the function is onto.Therefore,$f$ is one-to-one and onto.

Let the function $f:R \to R$ be defined by $f(x) = 2x + \sin x, x \in R$. Then $f$ is

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