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Thread: Parameters of Domain and Range: "For F(x) = 2/x, the domain cannot be 0."

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    Parameters of Domain and Range: "For F(x) = 2/x, the domain cannot be 0."

    I have watched the first three videos from this: https://www.khanacademy.org/math/alg...erval-notation

    I am confused as to all the rules that govern what the domain and range are of any particular function.

    So an example of a rule that I wouldn't know how to find in all situations is when the domain cannot be a specific number.

    Ex.:

    F(x)=2/x

    The domain cannot be 0. This is just one example of a parameter of what the domain cannot be.

    I guess my question is how do you know what the domain can be in all situations?

    Same thing applies to Range.

    So far, I have tried to think about the rules governing what constitutes as a parameter of the domain or range. I feel like there may be too many rules that govern what is a parameter of domain and range in relation to all the specific functions that are out there, so IDK how to go about knowing these rules that the parameters of the domain and range COULD fall under.

    I apologize if this is in the wrong section, mods can move it if it is not in the right sesion.
    Last edited by Quick; 01-24-2018 at 10:23 PM.

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    The domain of a function is just the set of all values that are legal inputs. That is, if you can evaluate the function for a given value of x, then that value is in the domain; if you can't, it isn't.

    So all you have to do is to walk through the process of evaluating the function, looking for things that can go wrong.

    In your example, you are dividing by x, which you can't do if x is zero; that is the only thing that can go wrong. So the domain is the set of all real numbers except zero.

    The only other situation you are likely to see (assuming you are doing basic algebra, and not trigonometry or logarithms, for example) is a square root. Any value of x that would make a radicand negative is not allowed.

    There are, indeed, many different functions; but for each new kind of function, you will learn what to check. It's sort of like something you find in the user manual when you "buy" a new kind of function. Just add that to your list! But be sure to "read the manual" (that is, pay close attention to the conditions given for a function).

    Of course, carrying out what I've described for more complicated functions is something that takes time to learn; I wouldn't try to take you through all the details until you have a problem that requires it. But when you ask, we can help you!

    Now, range is an entirely different thing. There are no rules for finding range; you have to use whatever technique fits a particular function, and there may not be any! At your (presumed) level, you may just be shown the graph of a function and required to read the range from that; graphing a function is the basic technique that always "works", but is not always possible to do exactly. This is essentially all that is done in your video.

    By the way, be careful with terminology. You said, "The domain cannot be 0." No, the domain is not a number, it is a set of numbers. The right thing to say is that the domain does not include 0, or that 0 is not in the domain.

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    You can think of a function as being like a box that takes an input value (x) and spits out an output value (y) corresponding to that input, according to some rule.

    The result is that a function creates what mathematicians call a "mapping" from one set of numbers (the domain) to another set of numbers (the range). The function takes your input number and "maps" it (meaning assigns it) to some output number. (EDIT: so to clarify, every number in the domain can be thought of as having an arrow connecting it to some corresponding number in the range: that is the mapping)

    The domain is the set of all numbers that actually lead somewhere. In other words, the numbers that, when input to the function, will actually produce a defined output value.

    In the case of f(x) = 2/x, the value 0 is not in the domain of the function, because it does not produce an output that is a defined real number (2/0 is undefined).

    The range is the set of all output numbers that you can actually get to by supplying some kind of input value. If a number is not in the range of a function, it's because there is NO input value that you can supply that will produce that number as an output.

    In this example, I think that 0 is also not in the range of the function, because you can never get there using any input. You can make 2/x arbitrarily close to zero by making x arbitrarily large, but you can never make it reach exactly 0.
    Last edited by j-astron; 01-24-2018 at 11:42 PM.

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    Thank you both for answering my question.

    Quote Originally Posted by Dr.Peterson View Post
    There are, indeed, many different functions; but for each new kind of function, you will learn what to check. It's sort of like something you find in the user manual when you "buy" a new kind of function. Just add that to your list! But be sure to "read the manual" (that is, pay close attention to the conditions given for a function).
    This is pretty much exactly what I think helps me understand this better.

    I got tripped up on the second video when they were talking about different examples of what was not included in the domain. The two examples they gave were:

    f(x)=2/x AND
    g(y)=square root of (y-6)

    And then I started to think that there were an awful lot of different conditions where a domain wasn't applicable (don't know if that is the right way to say it) and I wanted to know more about what domains couldn't be found.

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    Quote Originally Posted by Quick View Post
    The domain cannot be 0. This is just one example of a parameter of what the domain cannot be.
    Here are some notes about your wording.

    Instead of saying the domain cannot be 0, you ought to say that 0 cannot be in the domain. (The domain is a set of numbers. It doesn't make sense to say, "a set is zero" because zero is a number, not a set).

    That doesn't look like a correct use of the term 'parameter'. Are you thinking of something like, "This is just one example of how a value must be excluded from a domain"?

    In algebra, a parameter is a variable constant.

    Assuming that you've learned the difference between a variable and a constant (let us know, if you're not sure), then I can explain 'parameter' with the following example.

    There are an infinite number of quadratic polynomials. We can symbolically represent any arbitrary quadratic polynomial like this:

    Ax^2 + Bx + C

    We call the numbers A,B,C coefficients. These symbols represent constants (x is a variable). For example,

    27x^2 + 44x - 102

    The value of symbol x can vary, but the coefficients are fixed (constant).

    When I'm done working with this polynomial, I can move on to another.

    -15x^2 -37x + 455

    Again, the coefficients are constants.

    As we work with one polynomial after another (perhaps finishing one exercise and starting another), the coefficients vary. In other words, the constants A,B,C are changing from one case to the next. Variable constants like these are called parameters.

    Hence, when we symbolically represent any arbitrary quadratic polynomial, symbols A,B,C are parameters.



    my question is how do you know what the domain can be in all situations?
    By learning about different functions and what can go wrongly.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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    Quote Originally Posted by mmm4444bot View Post
    That doesn't look like a correct use of the term 'parameter'. Are you thinking of something like, "This is just one example of how a value must be excluded from a domain"?

    In algebra, a parameter is a variable constant.
    I didn't comment on the use of "parameter" because it is commonly used (outside of math) in a broader sense; see here:
    4. Usually, parameters. limits or boundaries; guidelines: the basic parameters of our foreign policy.
    5. characteristic or factor; aspect; element: a useful parameter for judging long-term success.
    But it is certainly worth discussing.

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    Quote Originally Posted by Dr.Peterson View Post
    I didn't comment on the use of "parameter" because it is commonly used (outside of math) in a broader sense
    Yup -- I figured Quick was using it that way. The common sense of the word still fits, in some math, but it seems off when saying, "Parameters of Domain and Range". Maybe that statement could work when referring to open, closed, and half-open sets? Not sure. I like telling students about "variable constants", heh.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

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    Quote Originally Posted by mmm4444bot View Post
    Yup -- I figured Quick was using it that way. The common sense of the word still fits, in some math, but it seems off when saying, "Parameters of Domain and Range". Maybe that statement could work when referring to open, closed, and half-open sets? Not sure. I like telling students about "variable constants", heh.
    I fully agree.

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