Question

Let $A =[-1,1]$ and $f : A \rightarrow$ 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 $f: A \rightarrow A, f(x)=x|x|$
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.