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Let A=R-{3},B=R-{1} " and " f:A rarr B d...

Let `A=R-{3},B=R-{1} " and " f:A rarr B` defined by `f(x)=(x-2)/(x-3)`. Is 'f' bijective? Give reasons.

Text Solution

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(i) To rest whether f is one-one
Let `x_(1), x_(2) in A` and let `f(x_(1)) = f(x_(2))`
`rArr (x_(1) - 2)/(x_(1) - 3) = (x_(2) - 2)/(x_(2) - 3)`
`rArr x_(1) x_(2) - 2x_(2) - 3x_(1) + 6 = x_(1)x_(2) - 3x_(2) - 2x_(1) + 6`
`rArr x_(1) = x_(2)`. Hence f is one-one.
(ii) To test whether f is onto
Let `y in B` and let y = f(x)
`rArr y = (x-2)/(x-3) rArr x = (3y - 2)/(y-1) in A`
Hence f is onto.
Thus f is one-one onto i.e., f is bijective.
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