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प्रश्न
If f (x) = `(4x + 3)/(6x - 4) , x ≠ 2/3`, show that fof (x) = x for all ` x ≠ 2/3` . Also, find the inverse of f.
उत्तर १
f (x) = `(4x + 3)/(6x - 4) `
`f (f (x)) = (4 f(x) + 3)/(6 f(x) - 4)`
`f(f(x))= (4 ((4x + 3)/(6x - 4))+3)/(6((4x + 3)/(6x - 4))-4)`
` fof (x) = (((16x + 12 + 18x - 12)/(6x -4)))/(((24x + 18 - 24 x + 16)/(6x - 4)))`
` fof (x) = (34x)/34`
fof (x) = x
For inversere y = `(4x + 3)/(6x - 4)`
6xy - 4y = 4x + 3
6 xy - 4x = 4y + 3
x(6y - 4) = 4y + 3
`x = (4y + 3)/(6y - 4) ⇒ y = (4x + 3)/(6x - 4)`
`⇒ f^(-1) (x) = (4x + 3)/(6x - 4)`
उत्तर २
`f(x) = (4x +3)/(6x -4) x ≠ 2/3`
`f "of"(x) = (4((4x +3)/(6x - 4))+ 3)/(6((4x +3)/(6x - 4)) - 4)`
= `(16x + 12 + 18x - 12)/(24x + 18 - 24x + 16)`
= `(34x)/(34) = x`
Therefore, fof (x) = x, for all `x ≠ 2/3`
⇒ fof = I
Hence, the given function f is invertible and the inverse of f is itself.
`y = (4x + 3)/(6x - 4)`
`6xy - 4y = 4x +3`
`6xy - 4y = 4y +3`
`x = (4y + 3)/(6y -4)`
∴ `f(x) = (4x +3)/(6x - 4)`
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