Is sentence "the equation is a function" the shorthand of saying "the equation defines a function"?
It's rather sloppy, though, to say "The equation is a function", because equations and functions are different things. A function, , say, is a relation between the elements of one set (the domain) and the elements of another set (the codomain) which has the property that every element in the domain is related with just one element in the codomain. If denotes an element in the domain, then is used to denote the element in the codomain that's related to it.
If we now introduce a second variable, , say, to stand for the element in the codomain that's related to , then we can write the equation:
and say that this equation "defines" the function. But it's not technically correct to say that this equation "is" the function.
However, it would also be correct to say that this equation also defines a straight line graph. But an equation and a graph are not the same thing. So it wouldn't technically be correct to say that the equation is a straight line, would it?
And, as I said, an equation and a function are not the same thing either. So, personally, I should try to avoid saying that an equation is a function, just as I should avoid saying that an equation is a straight line.
When we say that a "certain equation defines a function", would it also mean that this equation is used to constitute the function.
The same thing with the statement "the equation also defines a straight line graph." Does this also mean that the equation is used to constitute the straight line graph?
It is just worth stressing that, for a given , is unique. In other words, if and , then . This is what distinguishes a function from a more general relation, where we may have two different 's each related to the same .