{"id":818,"date":"2022-08-30T15:07:41","date_gmt":"2022-08-30T15:07:41","guid":{"rendered":"https:\/\/unknownerror.org\/index.php\/2013\/11\/09\/what-is-the-domain-and-range-of-y-xx-collection-of-common-programming-errors\/"},"modified":"2022-08-30T15:07:41","modified_gmt":"2022-08-30T15:07:41","slug":"what-is-the-domain-and-range-of-y-xx-collection-of-common-programming-errors","status":"publish","type":"post","link":"https:\/\/unknownerror.org\/index.php\/2022\/08\/30\/what-is-the-domain-and-range-of-y-xx-collection-of-common-programming-errors\/","title":{"rendered":"What is the domain and range of y = x^x?-Collection of common programming errors"},"content":{"rendered":"<li>This is one of my favorite functions. x can equal zero. Only when dealing with limits is 0\u2070 undefined, otherwise it is 1. x will only be defined in the negatives for integer values because you can&#8217;t take the root (fractional exponent) of a negative number. Since we&#8217;re working with real numbers, don&#8217;t worry about that. Otherwise x is defined for all real numbers, and the function will go off to infinity. The range is going to be the trickiest thing to find. At x = 1 it is obviously 1 and at x = 0 it is 1, but between them it will be smaller than 1 because we know that (1\/2)^(1\/2) = 1\/\u221a2 &lt; 1. Because y(0) = 1 and y(1) = 1 you can conclude by Rolle&#8217;s theorem (a specific case of the mean value theorem) that the derivative will equal zero somewhere between x = 0 and x = 1. This will be the minimum. To derive this you much use logarithmic differentiation. ln(y) = xln(x) (dy\/dx) \/ y = ln(x) + (x\/x) = ln(x) + 1 dy\/dx = y(ln(x) + 1) = (x^x)(ln(x) + 1) 0 = (x^x)(ln(x) + 1) x^x = 0 which doesn&#8217;t happen ln(x) + 1 = 0 ln(x) = -1 x = e\u207b\u00b9 = 1\/e y = (1\/e)^(1\/e) = (e\u207b\u00b9)^(e\u207b\u00b9) = e^(-e\u207b\u00b9) Domain: {x \u2208 \u211d | x \u2265 0} Range: {y \u2208 \u211d | y \u2265 e^(-e\u207b\u00b9)} If you want to include negative numbers and not complex numbers write it as Domain: {x \u2208 \u211d | x \u2265 0} \u222a {x \u2208 \u2124 | x &lt; 0} 0}\n<p>The range changes because as x approaches negative infinity the function approaches zero.<\/p>\n<\/li>\n<p id=\"rop\"><small>Originally posted 2013-11-09 22:47:20. <\/small><\/p>","protected":false},"excerpt":{"rendered":"<p>This is one of my favorite functions. x can equal zero. Only when dealing with limits is 0\u2070 undefined, otherwise it is 1. x will only be defined in the negatives for integer values because you can&#8217;t take the root (fractional exponent) of a negative number. Since we&#8217;re working with real numbers, don&#8217;t worry about [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-818","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/posts\/818","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/comments?post=818"}],"version-history":[{"count":0,"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/posts\/818\/revisions"}],"wp:attachment":[{"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/media?parent=818"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/categories?post=818"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/unknownerror.org\/index.php\/wp-json\/wp\/v2\/tags?post=818"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}