Does f mean the same thing as f(x) ?

1. Oct 29, 2007

lLovePhysics

Does "f" mean the same thing as "f(x)"??

I'm wondering if $$f$$ means the same thing as $$f(x)$$. Does $$f$$ refer more to the graph and f(x) refers to the y-coordinate of the graph?

Which one should I use to complete this sentence?:

___ is increasing on interval (-1,2)

or

___ has a relative minimum at x=0.

My book uses $$f$$ but my teacher uses $$f(x)$$ so I'm wondering if my teacher would mark me off if I used $$f$$. Probably not right?

2. Oct 29, 2007

lLovePhysics

Also, the minimums and maximums are y-values right? My teacher says that if the question asks "what is the min/max?" the answer should be given in the y-value. If the question asks "where is the min/max?" the answer should be given in the x-value.

3. Oct 29, 2007

neutrino

f is a "rule" of assigning values to every member of a set called the domain of f, to a unique member of another set called the co-domain of f. This "f" is a function. (Well, that was not a very precise definition, but it'll do for now.)

f(x) refers to the value that f takes at x. In other words, it is a single point.

I've seen some people (and some books) say something along the lines of "Let f(x) be a function...". But that's not right. It's f that is the function.

An extremum occurs at certain points where f' is zero. As the name suggests, a maximum (or minimum), is the greatest (or lowest) value that the function takes within a specific interval. Therefore it refers to f(x) (or y, if you prefer), where f'(x) = 0. Where this value occurs...that's at x.

Last edited: Oct 29, 2007
4. Oct 29, 2007

CompuChip

Also, questions often ask for the coordinate of the maximum or intersection, etc.
I would always try to write down the answer like
"The maximum/minimum/intersection/... is (y value) at x = (x value)"
then you always give enough information.

5. Oct 29, 2007

Gib Z

"...f which for every x value assigns the value x squared". =] I still don't think thats completely right, but you get my drift :P

6. Oct 29, 2007

HallsofIvy

Staff Emeritus
It would be correct to say "Let f be a function such that f(x)= x2". There are some mathematicians who advocate the form f( )= ( )2. I'm not quite ready for that!

Typo: "it refers to f(x) (or y, if you prefer)" NOT f'(x) or y'.

Strictly speaking "f" refers to the function itself (which does not necessarily have anything to do with a "graph") while f(x) refers to a specific value of the function. But the notation is "abused" frequently.

Strictly speaking, both of those should be f.

I wouldn't mark off for either but if you are concerned go with what your teacher uses- the author of your text book isn't grading you! (Or you could just ask your teacher!)

Last edited: Oct 29, 2007
7. Oct 29, 2007

Smartass

I disagree. The first one should be f and the second should be f(x), else minimum at x=0 would not make much sense (as f may not necessarily operate on x).

8. Oct 29, 2007

neutrino

Thank you very much, Halls. :)

9. Oct 29, 2007

HallsofIvy

Staff Emeritus
Gosh, I disagree with you! "f has a minimum at x= 0" makes sense because we had already been told that f is a function of x (i.e. we are using x to represent the independent variable). To say that "f(x) has a minimum at x=0" would not make sense because f(x) is a number (the value of f at some specific x) and it doesn't make sense to say that a number has a minimum!

(Pistols at thirty paces?)

10. Oct 29, 2007

Smartass

Where?
f(x) is a function in x, it is not a unique number, rather, it depends on x. So f(x) can be argued to mean that it is a number dependent on x, and this number attains a minimum at x=0.

Pistols at wha?

Last edited: Oct 29, 2007
11. Oct 29, 2007

NateTG

The notion of relative maximum depends on $f$ on $(-\epsilon,\epsilon)$, so it would probably be better to use $f$ there.

As with many other notations, it's common to play fast and loose, and rely on context. It's not so much of an issue with $f$ but the trailing parens are sometimes used to indicate that something is a function. This is popular in parametric notation with $x(t)$ and $y(t)$ or when denoting paths $\gamma(t)$.

Aside from context, there's really no good method to distinguish between $x(t)$ referring to a function and the same expression referring to a single value. $f(x)$ is less likely to refer to a function, but you're not likely to confuse anyone by using it that way as long as there is sufficient context. ($\gamma(t)$ by contrast is more likely to be a function.)

12. Oct 29, 2007

andytoh

f is the set of ordered pairs {(a,b),...}, where each a is in the domain of f and b is the image of a under the mapping f. In particular, if a=x, then b=f(x).