I stumbled upon this problem on composite functions in a school magazine under 'Quiz'. The problem says:
You are given the composite function $f(x+f(y))=bx+cy$ where $b$ and $c$ are real numbers, $b\neq-1$ with $x$ and $y$ being real variables.
Find the value of $f(\frac 1b)$ and the relationship between $b$ and $c$.
So far, I have worked out that:
$f(f(y)) = bx+cy-x$
$\therefore f(f(0))=bx-x$ ; calling this equation 1
$\therefore f(0)=cy-f(y)$ ; calling this equation 2
Substituting equation 2 into equation 1,
I get: $f(cy-f(y))=bx-x$
but this gets me to an equation similar to the original function.
Have my operations been correct? How do I continue from where I have left off to arrive at the desired answer?