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Question

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f: R ā†’ R defined by f(x) = 3 āˆ’ 4x

(ii) f: R ā†’ R defined by f(x) = 1 + x2

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Solution

(i) f: R ā†’ R is defined as f(x) = 3 āˆ’ 4x.

.

āˆ“ f is one-one.

For any real number (y) in R, there existsin R such that

āˆ“f is onto.

Hence, f is bijective.

(ii) f: R ā†’ R is defined as

.

.

āˆ“does not imply that

For instance,

āˆ“ f is not one-one.

Consider an element āˆ’2 in co-domain R.

It is seen thatis positive for all x āˆˆ R.

Thus, there does not exist any x in domain R such that f(x) = āˆ’2.

āˆ“ f is not onto.

Hence, f is neither one-one nor onto.


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