Is sentence "the equation is a function" the shorthand of saying "the equation defines a function"?
Hello mamenYes, I expect so, but I should like to see the whole context before being sure.
It's rather sloppy, though, to say "The equation is a function", because equations and functions are different things. A function, $\displaystyle f$, 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 $\displaystyle x$ denotes an element in the domain, then $\displaystyle f(x)$ is used to denote the element in the codomain that's related to it.
If we now introduce a second variable, $\displaystyle y$, say, to stand for the element in the codomain that's related to $\displaystyle x$, then we can write the equation:
$\displaystyle y = f(x)$and say that this equation "defines" the function. But it's not technically correct to say that this equation "is" the function.
Grandad
Hello mamenI agree with you - it does mean the equation $\displaystyle f(x) = x+2$ defines a 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 $\displaystyle f(x) = x+2$ 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.
Grandad
Lastly sir,
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?
thanks sir!
Hello mamenAgain, these are different English words - "expressed", "represented" - which have similar meanings to the ones you have already asked about. So, yes, you could say that an equation "expresses" a function, or "represents" a function, in just the same way as an equation "defines" a function.
Grandad
In my humble opinion a lot of math books consider functions to be a relation across the domain and codomain. In other words a function $\displaystyle f:X\mapsto Y$ is really just a set $\displaystyle R\subseteq X\times Y$ with the defining characterstic $\displaystyle R=\left\{(x,y)\in X\times Y:y=f(x)\right\}$ where $\displaystyle f(x)$ is the rule of correspondence.
Do you agree Grandad?
Hello Drexel28Yes. You have expressed in more formal notation the definition of a function that I gave in my first reply.
It is just worth stressing that, for a given $\displaystyle x$, $\displaystyle y$ is unique. In other words, if $\displaystyle y_1= f(x)$ and $\displaystyle y_2 = f(x)$, then $\displaystyle y_1 = y_2$. This is what distinguishes a function from a more general relation, where we may have two different $\displaystyle y$'s each related to the same $\displaystyle x$.
Grandad