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If F (X) = 4 X + 3 6 X − 4 , X ≠ 2 3 , Show that Fof (X) = X for All X ≠ 2 3 . Also, Find the Inverse of F. - Mathematics

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Sum

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.

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Solution 1

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)`

Solution 2

`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)`

Concept: Types of Relations
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