Difference between revisions of "Nonconstant"

 
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A polynomial <math>f(x)</math> is called '''nonconstant''' if <math>f(x)\not=a</math> for some fixed number <math>a</math>.
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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, 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 11:10, 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 $p(x) = x^2 - x + 1$ with the real numbers as domain and codomain is nonconstant. We can show this simply by noting that $p(1) = 1$ and $p(2) = 3$, so the function takes at least two different values. However, the function $f: \mathbb{Z} \to \mathbb{Z}$ such that $f(x) = 1$ for all $x$ is a constant function, and thus non-constant.

Note that recognizing non-constant functions is not always trivial. For example, the function $f: \mathbb{Z} \to \mathbb{Z}$ which takes an integer $x$, computes the value of $x^5 -2x^4 -2x^3 - x^2 + x + 4$ 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.

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