Oftentimes functions described by $f(x) = 2x+4$, and when this is mapped to the Cartesian plane, $f(x) = y$. This surely implies that $y = 2x+4$. Is there a difference between this and $y(x) = 2x+4$?
Notation – Difference Between y(x) and f(x)
notation
Related Solutions
Some authors like to distinguish the assertion "A is equal to B" from "define A as an object equal to B".
In math, the first one is always $A=B$ (I've never seen anything else, at least). But for definitions, I've seen
- $A \triangleq B$
- $A \equiv B$
- $A \stackrel{\mathrm{def}}{=} B$
- $A := B$
Often the triple equal sign is used for strong notions of equivalence (such as in binary relations) or with the "mod" symbol.
In computer programming, we are typographically limited of course. The assignment operator is usually distinguished from the comparison operator, and this is done in different ways depending on the language. For example in Pascal you assign x := 5
and compare x = 5
. However in C you assign x = 5
and compare x == 5
. (It should be noted that in C this syntax has caused untold confusion and a few famous bugs.)
My favorite (meaning "most awful") example is in PHP and javascript where one can do a "strict compare" with x === 5
. Finally, some languages have ridiculous vagaries like distinguishing which comparision operator is used based on the type of variable (usually strings being different from numerics), with syntaxes like .eq.
and related.
It really just varies according to the author/instructor. The only universal rule is that we use single bars for absolute values of real (and complex) numbers (e.g.$|-5|$). Once we start defining norms for other objects, we can choose single bars, double bars, or some other notation (although bars are very standard). In some contexts, we use $N(\alpha)$ to indicate a norm of $\alpha$. Reasons for using double bars (or any other notation) might include the desire to differentiate a vector norm, or some other norm, from the absolute value of a scalar.
If you are defining some kind of norm in your own writing, it's a good idea to define your notations before you use them, so that readers can follow your argument, even if they come from a context of using different notations.
(Single bars for absolute value is nearly universal. In some computer systems, however, absolute value of a real number $x$ is denoted $\mathrm{abs}(x)$. There may be other notations floating around, too.)
Best Answer
Functions vs. coordinates
Consider plots of two different functions: $f(x)$ and $g(x)$ on the same $xy$ plane. One curve will be labeled $y=f(x)$ which means "this is a set of $(x,y)$ points that satisfy $y=f(x)$ condition". The other will be labeled $y=g(x)$.
Which of these should define $y(x)$? Both? – certainly not, because $f$ and $g$ are different functions. I say: neither. The statement $y=f(x)$ is just a condition for some set of points (i.e. $(x,y)$ pairs) while $y=g(x)$ is another condition for another set of points.
Explicit definition in a form $y(x)=…$ does define a function (well, does or doesn't, read the next paragraph). In this case $y$ is just an arbitrary name and may replace $f$. The same symbol $y$ may be a coordinate on $xy$ plane, which was $xf$ plane before the name replacement. (It is only a custom to have $xy$ plane.) This "union" of function name and coordinate name may cause a problem when there is another function $g(x)$ to plot.
It should be obvious that if $y$ replaces $f$ it cannot replace $g$ that is different than $f$.
For that reason it is a good thing to have coordinates with symbols which are not function names.
Definitions vs. equations or conditions
Another problem: we often write function definitions the same way as conditions to be met or equations to solve. Compare the two:
$$cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} $$ $$cos(x) = \frac{1}{2}$$
The former may be treated as non-geometric definition of $cos$ function. The latter is just the equation to solve for $x$. We have some experience and often feel the difference, but a person (say: Bob) completely unaware of $cos$ will be confused. Bob may find every $x$ that satisfies
$$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = \frac{1}{2}$$
and still will not be able to tell what number $cos(x)$ is equal to for any other $x$.
It's worse than that! Bob cannot tell what number $cos(x)$ is equal to even for $x$ being his solution, because he cannot be sure that either equation defines the function (we know it's the first one, Bob doesn't). To clarify that, let's see what happens when I change $cos$ to $sin$ only:
$$sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} $$ $$sin(x) = \frac{1}{2}$$
We know by experience that neither of above defines $sin$. Yet these are legitimate equations to solve either separately or as a system (with empty set solution). Bob (not knowing about $sin$) may only assume that one of the equations is a definition – this will be wrong, his set of solutions will not be empty.
That's why I like the notation $f(x) \equiv …$ or the word "def" above the equality sign, or the explicit statement ("let us define…") – just to cut out possible ambiguity.
I've got the impression that you meant $y(x) \equiv 2x+4$ because there is no other expression in your example that you may want to define as $y(x)$.
Summary
Is there a difference between $y=2x+4$ and $y(x) \equiv 2x+4$?
Is there a difference between $y \equiv 2x+4$ and $y(x) \equiv 2x+4$?