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# Let $A=[-1,1].$ Then,dicuss whether the following functions defined on $A$ are one-one,onto or bijective:$\; f(x)\;=\;\frac{x}{2}$

Note:This is the 1st part of the 4 part question

Toolbox:
• A function $f: A \rightarrow B$ 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 $f(x) = y$.
• A function is bijective if it is both one-one and onto
Given $f(x)=\frac{x}{2}\qquad x \in [-1,1]$
Let $f(x)=f(y)$
Step1: Injective or One-One function:
=>$\frac{x}{2}=\frac{y}{2}$
=>$x=y$
f is one one
Step 2: Surjective or On-to function:
$f(x)=y=1$
$=>\frac{x}{2}=1$
$x=2$
$2 \in [-1,1]$
There does not exists an element n in A
Such that $f(x)=y\qquad \;for\;y=1$
Therefore f is not onto
Solution:Hence f is one-one but not onto

edited Mar 27, 2013 by meena.p