Find f(y) if y = (ax+b)/(cx-d)

Find \(f(y)\) if \(y=f(x)=\dfrac{ax+b}{cx-d}\)

Question

\[ y=f(x)=\frac{ax+b}{cx-d} \]

then find

\[ f(y) \]

Given

\[ y=\frac{ax+b}{cx-d} \]

Now,

\[ f(y)=\frac{ay+b}{cy-d} \]

Substitute the value of \(y\):

\[ f(y)= \frac{ a\left(\frac{ax+b}{cx-d}\right)+b }{ c\left(\frac{ax+b}{cx-d}\right)-d } \]

Simplify the numerator:

\[ = \frac{ \frac{a(ax+b)+b(cx-d)}{cx-d} }{ \frac{c(ax+b)-d(cx-d)}{cx-d} } \] \[ = \frac{ a^2x+ab+bcx-bd }{ acx+bc-cdx+d^2 } \]

Combine like terms:

\[ = \frac{ (a^2+bc)x+b(a-d) }{ (ac-cd)x+(bc+d^2) } \]

\[ \boxed{ f(y)= \frac{ (a^2+bc)x+b(a-d) }{ c(a-d)x+(bc+d^2) } } \]

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