Question

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

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

Answer 1

Answer: (D)

Given $A=[-1,1]$ and $f:A\to A,f(x)=x|x|$
Which implies $f(x)=x^2$ for $x>0$ and $f(x)=-x^2$ for $x<0$
When $x$ goes from $0$ to $1$ , the function $f(x)=x^2$ goes from $0$ to $1$ and when $x$ goes from $0$ to $-1$, the function $f(x)=-x^2$ goes from 0 to $-1$
So as $x$ moves from $-1$ to $1,f(x)$ also moves from $-1$ to $1$ as shown in the figure So, the given function is one-one and onto function.