What is the difference between an equation written in function notation and one that is not?

An equation is an equality which is satisfied by a unique set of values of your variables. You have, after the ##=## sign a fixed value, a fixed result.
For example: the equation ##4x-2=0## has zero as result and only ##x=1/2## as solution; this means that if you substitute the value of ##x=1/2## in the equation you have the result zero, i.e., the equation is satisfied.
Now, a function is similar, the only difference is that now you can have a lot of results after the ##=## sign and so you can have a lot of solutions.
For example: the function ##4x-2=y## doesn’t have a definite result (as before that was zero) but another variable ##y##, so every time you choose an ##x## you’ll get the corresponding value of ##y## that satisfies it.
If you choose:
##x=1 -> y=2##
##x=2 -> y=6##
….etc.
If ##x=1/2 -> y=0## which is the solution that we found before for our specific equation (in which the ##y## was already set as zero)!
So to summarize, an equation has a fixed result (after the ##=## sign) and an unique set of solutions (values of the variables); a function can have a lot of results (possibly ##oo##) and, as a consequence, a lot of solutions.
hope it helps

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