# What does this equation mean (approx with subscript)?

This might be a silly question, but I have no clue what it means: it's from this question i'm working on: I'm taking a CS course, and all the math i've done is calculus and linear algebra. I've never see approximation squiggly lines with a subscript. What on earth? I wish schools prepared us more properly, before throwing us the wolves. Half the time I'm spending trying to decipher what any of this means mathematically.

The text you quote defines exactly what $a \approx_f b$ means: $a \approx_f b$ if and only if $f(a) = f(b)$. For example, if we let $g: \mathbb R \rightarrow \mathbb R$ be defined by $g(x) = x^2$, then $2 \approx_g -2$ because $g(2) = 4 = g(-2)$, but it is not the case that $1 \approx_g 2$, because $g(1) = 1$ and $g(2) = 4$.
The passage you have there defines $\approx_f$ to be a certain relation. For any $x$ and $y$, $x \approx_f y$ is either true or false. The passage gives a definition for exactly when it is true.