# Question #2ae17

Aug 23, 2014

This is probably debatable. Expressions are somewhat implicit because you need to be told what to do with them. Functions on the other hand are very explicit; it tells you what it needs and it will give you something back.

Remember that math is a language so it has rules (a lot of them). "5" is an expression. So, what's the answer?

• positive?
• odd?
• prime?
• part of the Fibonacci sequence?
• the number before 6?
• half of 10?

So giving an expression without context can be meaningless.

You've seen this a thousand times: $y = {x}^{2} + 2 x + 7$. Is it an equation or a function? Again, it depends on the context. If this is all you are given, then you likely to treat this as a function, that is, $y \left(x\right) = {x}^{2} + 2 x + 7$. However, if I also give you $y = - {x}^{2} - 3 x + 20$ then you have a system of linear equations.

To be properly define a function, it should always be defined with the function name and its parameters:

$f \left(x\right) = 3 {x}^{4} - 5 {x}^{2}$
$\sin \left(\theta\right) = \ldots$
$g \left(x , y\right) = 3 \sin \left(x\right) + 4 \cos \left(y\right)$

Now, if we define something like:

$f \left(x\right) = \alpha {x}^{2} + \beta x + \gamma$

We know, that the only variable (the thing that we can change) is $x$. The rest are just constants like $\pi$. Maybe we know what they are, maybe we don't, but the function will give us a value back.

Yes, it's more work using function notation, but if everyone is clear about what's going on you can use the "=" sign.

I hope this is clear enough.