How do you tell the difference between function symmetry and reflections... and inverse.

Question

New learner here.

I have trouble telling the differences, by looking at graphs. Some functions looks like reflections but in fact are inverses.

Another example, reciprocal function; $f(x) = 1/x$.

Is it symmetrical or reflection? How do you tell? Any special technique, in general, you been using to tell them apart?

Thanks

Answer 1

Take any function $f(x)$ on a domain $[a,b]$.

Now, take it's reflection about the line $y=x$. The new curve that you get may or may not describe a function. If it is a function, let us call it $g(x)$.

If $g(x)$ exists, it is the inverse of $f(x)$ - can you see why? (Because $g(y) = f(x)$)

Now, it may so happen that for some special functions you start with, this reflection actually coincides with the original function - these functions are symmetric about $y=x$