Question

If $f\left(x\right)=x^3+3x^2+12x-2\sin x$ where $f:R\to R$, then prove that $f$ is bijective

Answer 1

A bijective function is a function which is onto and one-one,

Let's prove it onto first,

We can see that it's domain and range both is R, so it is onto.

Now let's prove it one-one,

For proving it one - one, it is enough to prove that function is strictly monotonic,

f'(x) =$3x^2+6x+12-2cosx>0$for all x $\in R$ So, f(x) is strictly monotonic in R.

Thus it is bijective.