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

Mar 9, 2015

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 $4 x - 2 = 0$ has zero as result and only $x = \frac{1}{2}$ as solution; this means that if you substitute the value of $x = \frac{1}{2}$ 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 $4 x - 2 = y$ 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:
$x = 1 \to y = 2$
$x = 2 \to y = 6$
....etc.

If $x = \frac{1}{2} \to y = 0$ 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 $\infty$) and, as a consequence, a lot of solutions.

hope it helps