SOLUTION OF PROBLEM: 10

Let function f is from R to R be defined by f(x) =2x + sin(x) for x belongs to R. Then f is

a) one-to-one and onto

b) one-to-one but not onto.

c) onto but not one-to-one

d) neither one-to-one nor onto.      ( DC/CH)

SOLUTION: The given function has co-domain as R( set of all real numbers)

For the function f(x) = 2x + sin(x) , 2x ranges from negative infinity to positive infinity whereas sin(x) varies from [-1,1] so ultimately the f(x) will have its range as R. Therefore, the f(x) is Onto function.

Also, f'(x) = 2 + cos(x) , cos(x) will vary from [-1,1], So 2+cos(x) varies from [1,3].

Therefore, f'(x) is always positive. Hence f(x) will always be an increasing function.Therefore, f(x) will be one-to-one.

So, the correct option is (a)