Question

 Let $f: A \rightarrow A$ where $A=\{x:-1 \leq x \leq 1\}$. Find whether the following functions are bijective.

(i) $x-\sin x$

(ii) $x|x|$

(iii) $\tan \frac{x}{4}$

(iv) $\mathrm{x}^4$

Answer 1

SOLUTION : $f:[-1,1] \rightarrow[-1,1]$

(i) $f(x)=x-\sin x \quad \text { (odd function) } \\$

$f^{\prime}(x)=1-\cos x \geq 0 \text { increasing function }$

Hence one - one

$f(-1)=-1+\sin 1 \\$

$f(1)=1-\sin 1$

Range $\equiv[-1+\sin 1,1-\sin 1]$

$\neq$ co domain function is not bijecive

(ii) $\quad f(x)=x|x|=\left\{\begin{array}{cc}x^2, & x \geq 0 \\ -x^2, & x<0\end{array}\right.$

one - one function

Range $\equiv[-1,1]=$ codomain

$\therefore \quad$ onto function

(iii) $f(x)=\tan \left(\frac{\pi x}{4}\right)$

by graph one-one onto

Bijective function

(iv) $f(x)=x^4$ even function

many-one $\Rightarrow \quad$ Not bijective