# 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

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.

#### 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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