Reasoning about a recursive function First of all, I am a computer science student, not a maths student. So maybe this is a trivial question, I just would like to understand it :)
Suppose I have the following (pointless) recursive function:
$$
f(x) = a + b \dot{} f(x)
$$
for some constants a and b. I suppose that it does not really compute any values, or maybe maps its argument to $\pm\infty$ (except for the cases $b=0$ or $a = 0, b = 1$). 
I would like to reason:
$$
f(x) - b \cdot f(x) = a \\
\vdots \\
f(x) = \frac{a}{1 - b} \\
$$
In terms of computation, this seems like a completely different function. So what is wrong about this reasoning (am I not allowed to treat the $=$ as equality? or $f(x)$ as a constant?).
 A: As pointed out in the comments, the issue you're running into is that your $f(x)$ isn't defined recursively.  When one says "$f$ is defined recursively," they mean that the value of $f(x)$ is a function of values of $f$ at points other than $x$.  For instance, the following are all recursive functions:
\begin{align}
f(x) &= a+b\cdot f(x-1),\quad x>0 \\
g(n) &= g(n-1) + g(n-2),\quad n\geq 2
\end{align}
(Notice that we need to specify the values of $x$ where the recurrence is valid; we also need some initial conditions or boundary values, but that strays from your current question.)
However, your function $f(x)$ does not depend on prior values of $x$.  Rather, you've really just described a constant function implicitly, rather than explicitly:
$$f(x)=a+b\cdot f(x) \implies (1-b)f(x) = a \implies f(x) = \frac{a}{1-b} $$
The equation on the far left is an implicit definition because you have defined $f(x)$ in terms of itself (not for values of $f(a),\;a\neq x$), while the equation on the right is a explicit definition.
