Question

Let $f:R\to R$ be defined by $f\left(x\right)=\cos (5x+2).$ Is f invertible?
if yes enter 1, else 0

Answer 1

No.For a function to be invertible it is necessary that it is bijective i.e. one-one and onto. Tbe funtion given here is neither surjective nor injective as shown below. Since$-1\le \cos (5x+2)\le 1,$ the range of $f=\{y:yisreal,-1\le y\le 1\}$ which is a proper subset of the co-domain R. Hence f is into so that it is not surjective, f is many-one since $\cos (5x+2)$ has the same value for many values of x. Thus $f(x+\frac{2}{5}n\pi )=\cos \{5(x+\frac{2}{5}n\pi )+2\}$ for all $n=0,\pm 1,\pm 2,\pm 3,....$ Since f is not bijective, it is not invertible.