Question:
If f(x)= 4x+3/6x-4 , x≠2/3 , show that fof(x)=x for all x≠2/3. what is the inverse of f?
Answer:
Given, f(x) = (4x + 3)/(6x - 4), x ≠ 2/3
Now, fof(x) = f{f(x)}
= f{(4x + 3)/(6x - 4)}
= {4(4x + 3)/(6x - 4) + 3}/{6(4x + 3)/(6x - 4) - 4}
= {4(4x + 3) + 3(6x - 4)}/{6(4x + 3) - 4(6x - 4)}
= {16x + 12 + 18x - 12}/{24x + 18 - 24x + 16}
= 34x/34
= x
So, fof(x) = x
Now, let y = (4x + 3)/(6x - 4)
=> y * (6x - 4) = 4x + 3
=> 6xy - 4y = 4x + 3
=> 6xy - 4y - 4x = 3
=> 6xy - 4x = 3 + 4y
=> x(6y - 4) = 4y + 3
=> x = (4y + 3)/(6y - 4)
So, f-1 (x) = (4x + 3)/(6x - 4)
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