Question

Let $A=[-1,1]$ and $f:A\to A$ be defined as $f\left(x\right)=x|x|$ for all $x\in A$, then $f\left(x\right)$ is

• A. many-one into function
• B. one-one into function
• C. many-one onto function
• D. one-one onto function

Answer 1

The correct option is D one-one onto function

Given $A=[-1,1]$ and $f:A\to A$ , $f\left(x\right)=x|x|$

Which implies $f\left(x\right)=x^2$ for $x>0$ and $f\left(x\right)=-x^2$ for $x<0$

When $x$ goes from $0$ to $1$, the function $f\left(x\right)=x^2$ goes from $0$ to $1$ and when $x$ goes from $0$ to $-1$, the function $f\left(x\right)=-x^2$ goes from $0$ to $-1$

So as $x$ moves from $-1$ to $1$ , $f\left(x\right)$ also moves from $-1$ to $1$ as shown in the figure

So, the given function is one-one and onto function.

Hence, option $D$ is correct.