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Let $X$ and $Y$ be subsets of $R$, the set of all real numbers. The function $f: X \rightarrow Y$ defined by $f(x)=x^2$ for $x \in X$ is one-one but not onto if (Here $R^{+}$is the set of all positive real numbers)
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$X=R^{+}, Y=R$
$\begin{array}{l} f\left(x_1\right)=f\left(x_2\right) \Rightarrow x_1^2=x_2^2 \Rightarrow x_1=x_2,\left[\text { if } X=R^{+}\right] \\ \Rightarrow f \text { is one-one. Since } R_f=R^{+} \subseteq R=Y ; \therefore f \text { is not onto }\end{array}$
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