https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Ruby5099991&feedformat=atom AoPS Wiki - User contributions [en] 2022-01-27T12:01:11Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Nonconstant&diff=25931 Nonconstant 2008-05-09T19:53:11Z <p>Ruby5099991: </p> <hr /> <div>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]] &lt;math&gt;p(x) = x^2 - x + 1&lt;/math&gt; with the [[real number]]s as [[domain]] and [[codomain]] is nonconstant. We can show this simply by noting that &lt;math&gt;p(1) = 1&lt;/math&gt; and &lt;math&gt;p(2) = 3&lt;/math&gt;, so the function takes at least two different values. However, the function &lt;math&gt;f: \mathbb{Z} \to \mathbb{Z}&lt;/math&gt; such that &lt;math&gt;f(x) = 1&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; is a [[constant]] function, as the co-domain of the function remains the same.<br /> <br /> Note that recognizing non-constant functions is not always trivial. For example, the function &lt;math&gt;f: \mathbb{Z} \to \mathbb{Z}&lt;/math&gt; which takes an integer &lt;math&gt;x&lt;/math&gt;, computes the value of &lt;math&gt;x^5 -2x^4 -2x^3 - x^2 + x + 4&lt;/math&gt; 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.</div> Ruby5099991 https://artofproblemsolving.com/wiki/index.php?title=Nonconstant&diff=25930 Nonconstant 2008-05-09T19:51:42Z <p>Ruby5099991: </p> <hr /> <div>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]] &lt;math&gt;p(x) = x^2 - x + 1&lt;/math&gt; with the [[real number]]s as [[domain]] and [[codomain]] is nonconstant. We can show this simply by noting that &lt;math&gt;p(1) = 1&lt;/math&gt; and &lt;math&gt;p(2) = 3&lt;/math&gt;, so the function takes at least two different values. However, the function &lt;math&gt;f: \mathbb{Z} \to \mathbb{Z}&lt;/math&gt; such that &lt;math&gt;f(x) = 1&lt;/math&gt; for all &lt;math&gt;x&lt;/math&gt; is a [[constant]] function, as the co-domain of the function remains the same regardless of changes to the domain.<br /> <br /> Note that recognizing non-constant functions is not always trivial. For example, the function &lt;math&gt;f: \mathbb{Z} \to \mathbb{Z}&lt;/math&gt; which takes an integer &lt;math&gt;x&lt;/math&gt;, computes the value of &lt;math&gt;x^5 -2x^4 -2x^3 - x^2 + x + 4&lt;/math&gt; 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.</div> Ruby5099991