If the function $f: R \rightarrow R$ is defined by $f(x)=x|x|$, then .

If the function $f: R \rightarrow R$ is defined by $f(x)=x|x|$, then .
  1. $f$ is one-one but not onto
  2. $f$ is onto but not one-one
  3. $f$ is both one-one and onto
  4. $f$ is neither one-one nor onto

Solution

Given, $f(x)=x|x|=\left\{\begin{array}{l}x(x), x \geq 0 \\ x(-x), x < 0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{l}x^2, x \geq 0 \\ -x^2, x < 0\end{array}\right.$, Also, $f: R \rightarrow R$
According to the graph, domain of $f(x)=$ Range of $f(x)=R$ $\therefore$ Range $=$ Codomain $=R$ $\Rightarrow f(x)$ is an onto function. If a line parallel to $X$-axis is drawn, it will intersect at only one point of $f(x)$. Hence, $f(x)$ is one-one also.

Asked in: AP EAMCET 2022 (08 Jul Shift 2)

Practice more Functions questions on Aicharya