Let f : R → R be the function defined by f (x) = sin (3x+2)∀x ∈R. Then f is invertible.

Question

State True or False for the statement:

Let f : R → R be the function defined by f (x) = sin (3x+2)∀x ∈R. Then f is invertible.

Answer 1

False

Given that, f : R → R be the function defined by

f (x) = sin (3x+2) ∀ x ∈R

f is invertible if it is bijective that is f should be one-one and onto.

Now, we know that sin x lies between -1 and 1.

So, the range of f(x) = sin (3x+2) is [-1,1] which is not equal to its co-domain.

i.e., range of f ≠ R (co-domain)

In other words, range of f is less than co-domain, i.e there are elements in co-domain which does not have any pre-image in domain.

so, f is not onto.

Hence, f is not invertible.