Please help me know the name of function:
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.
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 +6 +4 +81
(2 +3)^4 = (5)^4 = 625
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?
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.