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# if f(x)= 4x + 3 over 6x -4, x is not equal to 2 over 3, show that fof (x)=x, for all x not equal to 2 over 3. What is the inverse of f?

Q.4  If $f(x) = \frac{4x + 3}{6x - 4}, x \neq \frac{2}{3}$ show that $fof (x) = x$,  for all $x \neq\frac{2}{3}$. What is the inverse of $f$?

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$f(x) = \frac{4x + 3}{6x - 4}, x \neq \frac{2}{3}$

$fof (x) = x$

$(fof) (x) = f(f(x))$

$=f( \frac{4x + 3}{6x - 4})$

$=\frac{4( \frac{4x + 3}{6x - 4}) +3}{6( \frac{4x + 3}{6x - 4}) -4}$

$= \frac{16x+12+18x-12}{24x+1824x+16}$

$= \frac{34x}{34}$

$\therefore fof(x) = x$                ,  for all  $x \neq \frac{2}{3}$

$\Rightarrow fof=Ix$

Hence,the given function $f$is invertible and the inverse of $f$ is $f$ itself.

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