Question

The function $f:R\to R$ given by $f\left(x\right)=\cos x$ for all $x\in R$, is :

• A. surjective bu nor injective
• B. injective but nor surjective
• C. neither injective nor surjective
• D. injective

Answer 1

The correct option is C neither injective nor surjective

Injectivity : We know that $f\left(0\right)=\cos 0=1$ and $f\left(2\pi \right)=\cos 2\pi =1$.

So, different elements in R may have the same image. Hence, f is not an injection.

Surjectivity : Since the values of $\cos x$ lie between $-1$ and $1$, it follows that the range of $f\left(x\right)$ is not equal to its co-domain. So, f is not a surjection.