Let $A= {x:-1≤x≤ 1}=B$ be a function $f: A→B$. Then find the nature of following function.

$f(x)= x|x|$

bijective

$f: [-1, 1]-[1, 1]$, where $f(x) = x|x|$

Now $f(x)=x|x|=\left\{\begin{matrix}x×x,&x≥0\\x×(-x),&x<0\end{matrix}\right.=\left\{\begin{matrix}x^2,&x≥0\\-x^2,&x<0\end{matrix}\right.$

From the graph we observe that f(x) is one-one as there is no horizontal line which intersect the curve more than once. Also, we observe that the range is [-1, 1]. Hence f(x) is onto. Thus f(x) is bijective.