Let X and Y be subsets of R,the set of all re | vedclass

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(C) function $f: X 	o Y$ is one-one if $f(x_1) = f(x_2) \implies x_1 = x_2$ for all $x_1, x_2 \in X$.For $f(x) = x^2$,if $X = R^+$,then $x_1^2 = x_2^

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$X = R^+, Y = R$

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(C) function $f: X 	o Y$ is one-one if $f(x_1) = f(x_2) \implies x_1 = x_2$ for all $x_1, x_2 \in X$.For $f(x) = x^2$,if $X = R^+$,then $x_1^2 = x_2^2 \implies x_1 = x_2$ (since $x_1, x_2 > 0$),so $f$ is one-one.$A$ function is onto if the range of $f$ equals the codomain $Y$.Here,the range of $f$ is $R^+$. If $Y = R$,then the range $R^+ \subset R$,so $f$ is not onto.Thus,for $X = R^+$ and $Y = R$,the function is one-one but not onto.

Let $X$ and $Y$ be subsets of $R$,the set of all real numbers. The function $f:X \to Y$ 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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