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Homework Help: Does f mean the same thing as f(x) ?

  1. Oct 29, 2007 #1
    Does "f" mean the same thing as "f(x)"??

    I'm wondering if [tex]f[/tex] means the same thing as [tex]f(x)[/tex]. Does [tex]f[/tex] 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)


    ___ has a relative minimum at x=0.

    My book uses [tex]f[/tex] but my teacher uses [tex]f(x)[/tex] so I'm wondering if my teacher would mark me off if I used [tex]f[/tex]. Probably not right?
  2. jcsd
  3. Oct 29, 2007 #2
    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.
  4. Oct 29, 2007 #3
    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
  5. Oct 29, 2007 #4


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    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.
  6. Oct 29, 2007 #5

    Gib Z

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    "...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
  7. Oct 29, 2007 #6


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    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 by a moderator: Oct 29, 2007
  8. Oct 29, 2007 #7
    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).
  9. Oct 29, 2007 #8
    Thank you very much, Halls. :)
  10. Oct 29, 2007 #9


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    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?)
  11. Oct 29, 2007 #10
    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
  12. Oct 29, 2007 #11


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    The notion of relative maximum depends on [itex]f[/itex] on [itex](-\epsilon,\epsilon)[/itex], so it would probably be better to use [itex]f[/itex] 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 [itex]f[/itex] but the trailing parens are sometimes used to indicate that something is a function. This is popular in parametric notation with [itex]x(t)[/itex] and [itex]y(t)[/itex] or when denoting paths [itex]\gamma(t)[/itex].

    Aside from context, there's really no good method to distinguish between [itex]x(t)[/itex] referring to a function and the same expression referring to a single value. [itex]f(x)[/itex] 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. ([itex]\gamma(t)[/itex] by contrast is more likely to be a function.)
  13. Oct 29, 2007 #12
    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).
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