Question

# If $f:R \to R$, then $f\left( x \right) = x\left| x \right|$ will be A. many-one-ontoB. one-one-ontoC. many-one-intoD. one-one-into

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Hint: Here $R$ means a set of real numbers. The function $f:R \to R$ implies that the domain of the given function is $R$and corresponding range is also $R$. In simple words, if you put real numbers in the function you will get a set of real numbers. Plot the graph for the given function. So, use this concept to reach the solution of the problem.

Given, $f:R \to R$, $f\left( x \right) = x\left| x \right|$
$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} { - {x^2},}&{x < 0} \\ {0,}&{x = 0} \\ {{x^2}}&{x > 0} \end{array}} \right\}$
If we plot the graph for $f\left( x \right)$, it will be
Here the function $f:R \to R$ is bijective since its graph meets every horizontal and vertical line exactly once.
Since, the graph of $f\left( x \right)$ shows that it is a bijective (one-one-onto) function.