# Difference between revisions of "Nonconstant"

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− | A polynomial <math> | + | 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, and thus non-constant. |

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+ | 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 12:10, 29 January 2007

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, and thus non-constant.

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.