Difference between revisions of "Nonconstant"
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| − | A [[function]] is called '''nonconstant''' if it takes more than one value (if there is more than one element in its [[range]]). For example, the [[polynomial]] <math>p(x) = x^2 - x + 1</math> with the [[real number]]s as [[domain]] and [[codomain]] is nonconstant. We can show this simply by noting that <math>p(1) = 1</math> and <math>p(2) = 3</math>, so the function takes at least two different values. However, the function <math>f: \mathbb{Z} \to \mathbb{Z}</math> such that <math>f(x) = 1</math> for all <math>x</math> is a [[constant]] function, | + | A [[function]] is called '''nonconstant''' if it takes more than one value (if there is more than one element in its [[range]]). For example, the [[polynomial]] <math>p(x) = x^2 - x + 1</math> with the [[real number]]s as [[domain]] and [[codomain]] is nonconstant. We can show this simply by noting that <math>p(1) = 1</math> and <math>p(2) = 3</math>, so the function takes at least two different values. However, the function <math>f: \mathbb{Z} \to \mathbb{Z}</math> such that <math>f(x) = 1</math> for all <math>x</math> is a [[constant]] function, as the co-domain of the function remains the same regardless of changes to the domain. |
Note that recognizing non-constant functions is not always trivial. For example, the function <math>f: \mathbb{Z} \to \mathbb{Z}</math> which takes an integer <math>x</math>, computes the value of <math>x^5 -2x^4 -2x^3 - x^2 + x + 4</math> and then takes the [[remainder]] of this number on division by 3 appears quite complicated but turns out to be identical to the last function in the previous paragraph: it only takes the value 1. | Note that recognizing non-constant functions is not always trivial. For example, the function <math>f: \mathbb{Z} \to \mathbb{Z}</math> which takes an integer <math>x</math>, computes the value of <math>x^5 -2x^4 -2x^3 - x^2 + x + 4</math> and then takes the [[remainder]] of this number on division by 3 appears quite complicated but turns out to be identical to the last function in the previous paragraph: it only takes the value 1. | ||
Revision as of 15:51, 9 May 2008
A function is called nonconstant if it takes more than one value (if there is more than one element in its range). For example, the polynomial 
with the real numbers as domain and codomain is nonconstant. We can show this simply by noting that 
and 
, so the function takes at least two different values. However, the function 
such that 
for all 
is a constant function, as the co-domain of the function remains the same regardless of changes to the domain.
Note that recognizing non-constant functions is not always trivial. For example, the function which takes an integer , computes the value of 
and then takes the remainder of this number on division by 3 appears quite complicated but turns out to be identical to the last function in the previous paragraph: it only takes the value 1.
