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Let \(f : R \to R\) be defined as \(f (x) = 3x\). Choose the correct answer:

\begin{array}{1 1}(a)\;f\;is\;one-one\;onto. & (b)\;f\;is\;many-one\;onto.\\(c) \;f\;is\;one-one\;but\;not\;onto. & (d)\;f\;is\;neither\;one-one\;nor\; onto.\end{array}

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  • A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function.
  • A function$ f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
Given $f:R \to R$ defined as $f(x)=3x$
Let there be two elements $x,y \in R$ such that
Step1: Injective or One-One function:
Therefore $f:R \to R$ defined as $f(x)=3x$ is one-one since $ f(x)=f(y) =>x=y$
Step 2: Surjective or On-to function:
For any element y in codomain R. then exists $x=y/3$ in A such that $f(x)=y$
$f:R \to R$ defined as $f(x)=3x$ is onto since ,A function $f:A \to B $ is onto if for $y \in B \; then\;exist \; x \in A$ such that $f(x)=y$
$f:R \to R$ defined as $f(x)=3x$ is both one one and onto
Solution:'A' option is correct


answered Mar 14, 2013 by thagee.vedartham
edited Mar 20, 2013 by meena.p

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