Let the function f:R→R be defined by f(x)=2x+sinx for x∈R. Then f is
one-to-one and onto
one-to-one but not onto
onto but not one-to-one
neither one-to-one nor onto
Given that f(x)=2x+sinx,x∈R
or f′(x)=2+cosx
But −1≤cosx≤1
or 1≤2+cosx≤3
∴ f′(x)>0∀x∈R
Therefore, f(x) is strictly increasing and, hence, one-one.
Also, as x→∞,f(x)→∞ , and x→−∞,f(x)→−∞ . Therefore,
Hence, f(x) is onto.
Thus, f(x) is one-one and onto.