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Home » Let     be defined by     . Then show that f is bijective.
Let

    \[\mathbf{A}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{3} \right\},\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{1} \right\}.\text{ }\mathbf{Let}\text{ }\mathbf{f}\text{ }:\text{ }\mathbf{A}\text{ }\to \text{ }\mathbf{B}\]

be defined by

    \[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}/\text{ }\mathbf{x}\text{ }\text{ }\mathbf{3}\forall \mathbf{x}\in \mathbf{A}\]

. Then show that f is bijective.
CBSE | Class 12 | Exercise 1 | Maths | NCERT Exemplar | Relations and Functions
Let

    \[\mathbf{A}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{3} \right\},\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{1} \right\}.\text{ }\mathbf{Let}\text{ }\mathbf{f}\text{ }:\text{ }\mathbf{A}\text{ }\to \text{ }\mathbf{B}\]

be defined by

    \[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}/\text{ }\mathbf{x}\text{ }\text{ }\mathbf{3}\forall \mathbf{x}\in \mathbf{A}\]

. Then show that f is bijective.

According to the question,

    \[A\text{ }=\text{ }R\text{ }\text{ }\left\{ 3 \right\},\text{ }B\text{ }=\text{ }R\text{ }\text{ }\left\{ 1 \right\}\]

And,

f : A → B be defined by

    \[f\text{ }\left( x \right)\text{ }=\text{ }x\text{ }\text{ }2/\text{ }x\text{ }\text{ }3\forall x\in A\]

Hence,

    \[f\text{ }\left( x \right)\text{ }=\text{ }\left( x\text{ }\text{ }3\text{ }+\text{ }1 \right)/\text{ }\left( x\text{ }\text{ }3 \right)\text{ }=\text{ }1\text{ }+\text{ }1/\text{ }\left( x\text{ }\text{ }3 \right)\]

Let

    \[f({{x}_{1}})\text{ }=\text{ }f\text{ }({{x}_{2}})\]

So, f (x) is an injective function.

Now let

    \[y\text{ }=\text{ }\left( x\text{ }\text{ }2 \right)/\text{ }\left( x\text{ }-3 \right)\]

    \[\begin{array}{*{35}{l}} x\text{ }\text{ }2\text{ }=\text{ }xy\text{ }\text{ }3y \\ x\left( 1\text{ }\text{ }y \right)\text{ }=\text{ }2\text{ }\text{ }3y \\ x\text{ }=\text{ }\left( 3y\text{ }\text{ }2 \right)/\text{ }\left( y\text{ }\text{ }1 \right) \\ y\in R\text{ }\text{ }\left\{ 1 \right\}\text{ }=\text{ }B \\ \end{array}\]

Thus, f (x) is onto or subjective.

Therefore, f(x) is a bijective function.

i

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