Show that \(x=f(y)\)

Solution

Given: $$ y=\frac{ax-b}{bx-a} $$

Now, $$ f(y)=\frac{ay-b}{by-a} $$

Substitute the value of \(y\):

$$ f(y)= \frac{ a\left(\frac{ax-b}{bx-a}\right)-b }{ b\left(\frac{ax-b}{bx-a}\right)-a } $$

Taking LCM in numerator and denominator:

$$ = \frac{ \frac{a(ax-b)-b(bx-a)}{bx-a} }{ \frac{b(ax-b)-a(bx-a)}{bx-a} } $$

$$ = \frac{ a^2x-ab-b^2x+ab }{ abx-b^2-a^2bx+a^2 } $$

$$ = \frac{ x(a^2-b^2) }{ a^2-b^2 } $$

$$ =x $$

Therefore, $$ f(y)=x $$

Hence, $$ \boxed{x=f(y)} $$