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Math Help - What is the name of this function?

  1. #1
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    What is the name of this function?

    Please help me know the name of function:
    y=(ax+b)^r
    where "a", "b", and "r" are real constants and "r>0".
    It is clear that for "r=1", this equation is general form of a linear function; and also for positive integer values of "r" (i.e. Natural numbers), this equation is a polynomial function. But I don't know whether there is a special function name for positive real values of "r" or not. Any reply is appreciated.
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  2. #2
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    Quote Originally Posted by mottaghi View Post
    Please help me know the name of function:
    y=(ax+b)^r
    where "a", "b", and "r" are real constants and "r>0".
    It is clear that for "r=1", this equation is general form of a linear function; and also for positive integer values of "r" (i.e. Natural numbers), this equation is a polynomial function. But I don't know whether there is a special function name for positive real values of "r" or not. Any reply is appreciated.
    When r is a positive integer you have a polynomial functions.

    When r is not, then I do not know what you have.
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  3. #3
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    Quote Originally Posted by mottaghi View Post
    Please help me know the name of function:
    y=(ax+b)^r
    where "a", "b", and "r" are real constants and "r>0".
    It is clear that for "r=1", this equation is general form of a linear function; and also for positive integer values of "r" (i.e. Natural numbers), this equation is a polynomial function. But I don't know whether there is a special function name for positive real values of "r" or not. Any reply is appreciated.
    It is a binomial function. See Binomial expansion somewhere.
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    Quote Originally Posted by ticbol View Post
    It is a binomial function. See Binomial expansion somewhere.

    Thank you very much for your contribution.
    But I think binomial is an expresion of the form (a+bx) or (1+x) and so on which have two nomials, and not of the form (a+bx)^r. For any power of "r", the binomial " (a+bx) " can be expanded using binomial series...
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  5. #5
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    Quote Originally Posted by mottaghi View Post
    Thank you very much for your contribution.
    But I think binomial is an expresion of the form (a+bx) or (1+x) and so on which have two nomials, and not of the form (a+bx)^r. For any power of "r", the binomial " (a+bx) " can be expanded using binomial series...
    (x+y)^r
    (33x -0.2y)^r
    They are binomial functions too.
    Any two terms inside the parentheses, be they variables or constants, can be called a binomial "function".

    (2 +3)^4 = 2^4 +4[2^3 *3] +6[2^2 *3^2] +4[2 *3^3] +3^4
    = 16 + 4[24] +6[36] +4[54] +81
    = 625

    (2 +3)^4 = (5)^4 = 625
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  6. #6
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    Quote Originally Posted by ticbol View Post
    (x+y)^r
    (33x -0.2y)^r
    They are binomial functions too.
    Any two terms inside the parentheses, be they variables or constants, can be called a binomial "function".

    (2 +3)^4 = 2^4 +4[2^3 *3] +6[2^2 *3^2] +4[2 *3^3] +3^4
    = 16 + 4[24] +6[36] +4[54] +81
    = 625

    (2 +3)^4 = (5)^4 = 625

    Dear "ticbol"
    Thank you again. Knowing that:
    (x+y)^2 = x^2 + 2xy + y^2 ,
    do you mean the collected form is a binomial, while the expanded form is a polynomial (because has more than two terms)? On the other words, do you mean that polynomial function is a sub_set of binomial function?
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  7. #7
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    Quote Originally Posted by mottaghi View Post
    Dear "ticbol"
    Thank you again. Knowing that:
    (x+y)^2 = x^2 + 2xy + y^2 ,
    do you mean the collected form is a binomial, while the expanded form is a polynomial (because has more than two terms)? On the other words, do you mean that polynomial function is a sub_set of binomial function?
    I don't know about the polynomial function is a subset of binomial function. For sure (ax +by +cz)^r will produce polynomial functions too eventhough (ax +by +cz)^r is not a binomial function.

    The (x+y)^2 is a binomial function.
    It's expansion x^2 +2xy +y^2 is a polynomial function.
    Is the polynomial function a subset of the original binomial function? I don't know.
    What I'm saying, your y = (ax +b)^r can be called a binomial function in that it obeys a binomial expansion.

    To confuse you more, , say,
    y = (x^2 -5x +6) / (x -3)
    That is a rational or "fractional" function.
    Simplify or "expand" that,
    y = [(x-3)(x-2)]/(x-3)
    y = x-2
    The simplified form now is not "fractional" anymore.
    The original is not the same as the "expanded" or simplified form.
    So, back to your (x+y)^2, a binomial, .....blah, blah....

    -----------------------
    Ah, the (ax +by +cz)^r can also be binomial if it is first treated as [ax +(by +cz)]^r. That's only one of the three possible forms. [(ax +by) +cz]^r . [(ax +cz) +by]^r. Play with the binomial expansions of the 3 forms and you should get the same result in the end.
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