Question

Which of the following functions from $A=\left\{x\in R:-1\le x\le 1\right\}$ to itself are bijections?

(a) $f\left(x\right)=\left|x\right|$
(b) $f\left(x\right)=\sin \frac{\pi x}{2}$
(c) $f\left(x\right)=\sin \frac{\pi x}{4}$
(d) None of these

Answer 1

(b) $f\left(x\right)=\sin \frac{\pi x}{2}$

It is clear that f(x) is one-one.

$\text{Range of}f=\left[\sin \frac{\mathrm{\pi }\left(-1\right)}{2},\sin \frac{\mathrm{\pi }\left(1\right)}{2}\right]=\left[\sin \frac{-\mathrm{\pi }}{2},\sin \frac{\mathrm{\pi }}{2}\right]=\left[-1,1\right]=A=\text{Co domain of}f$
⇒ f is onto.
So, f is a bijection.