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   Functions: Domain and Range | Purplemath
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class="section-container main-container sidebar-both-container"><div class="section main-inner sidebar-both-inner"><div class=pad-grid><div class="col-custom content sidebar-both-content"><h1 role=main>Functions: Domain and Range</h1><div class="w3-container hide-print"style="padding:20px 0;"><p style=text-align:center;><a class=w3-btn href=fcns.htm style=background-color:#f1f1f1;> Functions v. Relations </a> <button style="color:#000; background-color:#f1f1f1;"class=w3-btn disabled>Domain & Range</button></div><div class=grid><div class="col-half tablet-col-half"><h2 class="section-title hide-print">Purplemath</h2><p class=text id=lesson><a name=return> </a> Let's return to the subject of domains and ranges.<p class=text>When functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. These won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. Small sets of points are generally the simplest sorts of relations, so your book starts with those.<p class=text>For instance:<div class=hide-print><p class="display-mobile content-continues"style=padding-top:20px;>Content Continues Below</div></div><div class="col-half tablet-col-half hide-print"><div id=mh_callout><h2 class=section-title-2>MathHelp.com</h2><div align=center id=mh_lesson_display><a href=https://www.mathhelp.com/how_to/functions/domain_and_range/?id=2037&utm_campaign=purplemath&utm_source=fcns2.htm&utm_medium=banner> <img alt="Domain and Range on MathHelp.com"height=200 src=https://www.mathhelp.com/images/lesson-detail/screenshots/285x200-play/2037-gui.jpg width=285> </a></div><div id=mh_notice><p style=text-align:center;><a href=https://www.mathhelp.com/how_to/functions/domain_and_range/?id=2037&utm_campaign=purplemath&utm_source=fcns2.htm&utm_medium=banner style=text-decoration:none;> Domain and Range </a></div></div></div></div><ul><li><h4 class="text question">State the domain and range of the following relation. Is the relation a function?</h4></ul><div style=margin-left:40px;><h4 class="math question">{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}</h4></div><div style="margin-left:20px; padding-top:20px;"><p class=text>The above list of points, being a relationship between certain <span class=math> <em> x </em> </span> 's and certain <span class=math> <em> y </em> </span> 's, is a relation. The domain is all the <span class=math> <em> x </em> </span> -values, and the range is all the <span class=math> <em> y </em> </span> -values. To give the domain and the range, I just list the values without duplication:<p class="text answer">domain: <span class=math> {2, 3, 4, 6} </span><p class="text answer">range: <span class=math> {–3, –1, 3, 6} </span></div><div class="alignright display-mobile nomargin hide-print"style=width:180px;><p class="in-lesson-resp-ad-header nomargin"align=center>Affiliate<div style="text-align:center; margin:10px 0px 10px 15px;"id=contract_ad><ins data-revive-id=22fb12d5af67c91fb385a68f631b6803 data-revive-zoneid=2> </ins><script src=//www.purplemath.com/mathads/www/delivery/asyncjs.php></script></div></div><p class=text>(It is customary to list these values in numerical order, but it is <em> not </em> required. Sets are called "unordered lists", so you can list the numbers in any order you feel like. Just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)<div class=indent_prob><p class=text>While the given set does indeed represent a <em> relation </em> (because <span class=math> <em> x </em> </span> 's and <span class=math> <em> y </em> </span> 's are being related to each other), the set they gave me contains two points with the same <span class=math> <em> x </em> </span> -value: <span class=math> (2, –3) </span> and <span class=math> (2, 3) </span> . Since <span class=math> <em> x </em> = 2 </span> gives me two possible destinations (that is, two possible <span class=math> <em> y </em> </span> -values), then <span class=answer> this relation is not a function. </span></div><p class=text>Note that all I had to do to check whether the relation was a function was to look for duplicate <span class=math> <em> x </em> </span> -values. If you find any duplicate <span class=math> <em> x </em> </span> -values, then the different <span class=math> <em> y </em> </span> -values mean that you do not have a function. Remember: For a relation to be a function, each <em> x </em> -value has to go to one, <em> and only one </em> , <em> y </em> -value.<ul><li><h4 class="text question">State the domain and range of the following relation. Is the relation a function?</h4></ul><div style=margin-left:40px;><h4 class="math question">{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}</h4></div><div style="margin-left:20px; padding-top:20px;"><p class=text>I'll just list the <span class=math> <em> x </em> </span> -values for the domain and the <span class=math> <em> y </em> </span> -values for the range:<div class=indent_prob><p class="text answer">domain: <span class=math> {–3, –2, –1, 0, 1, 2} </span><p class="text answer">range: <span class=math> {5} </span></div><p class=text>This is another example of a "boring" function, just like the <a href=fcns.htm#boring> example </a> on the previous page: every last <span class=math> <em> x </em> </span> -value goes to the exact same <span class=math> <em> y </em> </span> -value. But each <span class=math> <em> x </em> </span> -value is different, so, while boring,<div class=indent_prob><p class="text answer">this relation is indeed a function.</div></div><p class=text>In point of fact, these points lie on the horizontal line <span class=math> <em> y </em> = 5 </span> .<hr><p class=text>By the way, the name for a set with only one element in it, like the "range" set above, is "singleton". So the range could also be stated as "the singleton of <span class=math> 5 </span> "<hr><div class=hide-print><p class=in-lesson-resp-ad-header>Content Continues Below<div class="in-lesson-resp-ad align-center"><div id=Media.net></div><script data-cookieconsent=marketing type=text/plain>
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            </script></div><hr></div><p class=text>There is one other case for finding the domain and range of functions. They will give you a function and ask you to find the domain (and maybe the range, too). I have only ever seen (or can even think of) two things at this stage in your mathematical career that you'll have to check in order to determine the domain of the function they'll give you, and those two things are denominators and square roots.<ul><li><h4 class="text question">Determine the domain and range of the given function:</h4></ul><div id=domain04 style=margin-left:40px;></div><script>katex.render("\\mathbf{\\color{green}{\\mathit{y} = \\dfrac{\\mathit{x}^2 + \\mathit{x} - 2}{\\mathit{x}^2 - \\mathit{x} - 2}}}", domain04);</script><noscript><p><img alt="y = (x^2 + x - 2) / (x^2 - x - 2)"height=41 src=fcns/domain04.gif width=94></p></noscript><div style="margin-left:20px; padding-top:20px;"><p class=text>The domain is all the values that <span class=math> <em> x </em> </span> is allowed to take on. The only problem I have with this function is that I need to be careful not to divide by zero. So the only values that <span class=math> <em> x </em> </span> can not take on are those which would cause division by zero. So I'll set the denominator equal to zero and solve; my domain will be everything else.<div class=indent_prob><p class=math><em> x </em> <sup> 2 </sup> – <em> x </em> – 2 = 0<p class=math>( <em> x </em> – 2)( <em> x </em> + 1) = 0<p class=math><em> x </em> = 2 or <em> x </em> = –1</div><p class=text>Then <span class=answer> the domain is "all <span class=math> <em> x </em> </span> not equal to <span class=math> –1 </span> or <span class=math> 2 </span> ". </span><p class=text>The range is a bit trickier, which is why they may not ask for it. In general, though, they'll want you to graph the function and find the range from the picture. In this case:<div class="section img-scroller"><img alt=graph height=191 src=fcns/domain01.gif style=min-width:342px; width=342></div><p class=text>As you can see from my picture, the graph "covers" all <span class=math> <em> y </em> </span> -values; that is, the graph will go as low as I like, and will also go as high as I like. For any point on the <span class=math> <em> y </em> </span> -axis, no matter how high up or low down, I can go from that point either to the right or to the left and, eventually, I'll cross the graph. Since the graph will eventually cover all possible values of <span class=math> <em> y </em> </span> , then:<div class=indent_prob><p class="text answer">the range is "all real numbers".</div></div><ul><li><h4 class="text question">Determine the domain and range of the given function:</h4></ul><div style=margin-left:40px;><div id=domain05></div><script>katex.render("\\mathbf{\\color{green}{\\mathit{y} = -\\sqrt{-2\\mathit{x} + 3}}}", domain05);</script><noscript><p><img alt="y = -sqrt(-2x + 3)"height=19 src=fcns/domain05.gif width=98></p></noscript></div><div class="hide-mobile hide-print"style="width:300px; min-height:270px; float:right; margin-left:10px;"><p class=in-lesson-resp-ad-header>Advertisement</p><ins data-revive-id=22fb12d5af67c91fb385a68f631b6803 data-revive-zoneid=9> </ins><script src=//www.purplemath.com/mathads/www/delivery/asyncjs.php></script></div><div class=indent_prob><p class=text style=padding-top:20px;>The domain is all values that <span class=math> <em> x </em> </span> can take on. The only problem I have with this function is that I cannot have a negative inside the square root. So I'll set the insides greater-than-or-equal-to zero, and solve. The result will be my domain:<div class=indent_prob><p class=math>–2 <em> x </em> + 3 ≥ 0<p class=math>–2 <em> x </em> ≥ –3<p class=math>2 <em> x </em> ≤ 3<p class=math><em> x </em> ≤ 3/2 = 1.5</div><p class=text>Then <span class=answer> the domain is "all <span class="math answer"> <em> x </em> ≤ 3/2 </span> ". </span><p class=text>The range requires a graph. I need to be careful when <a href=graphrad.htm> graphing radicals </a> :<div class="section img-scroller"><img alt=graph height=193 src=fcns/domain02.gif style=min-width:328px; width=328></div><p class=text>The graph starts at <span class=math> <em> y </em> = 0 </span> and goes down (heading to the left) from there. While the graph goes down very slowly, I know that, eventually, I can go as low as I like (by picking an <span class=math> <em> x </em> </span> that is sufficiently big). Also, from my experience with graphing, I know that the graph will never start coming back up. Then:<div class=indent_prob><p class="text answer">the range is "all <span class=math> <em> y </em> ≤ 0 </span> ".</div></div><div class="hide-print hide-mobile"><p class=in-lesson-resp-ad-header>Affiliate<div class=in-lesson-resp-ad><a href=https://www.wyzant.com/algebra_1_tutors.aspx?utm_source=partners&utm_medium=displayad&utm_term=26&utm_campaign=17&g=3JY> <img alt="Algebra Tutors"style="display:block; margin:auto;"height=60 src=//data.wyzant.com/content/images/displayads/Algebra-468x60_new.jpg width=468> </a></div></div><ul><li><h4 class="text question">Determine the domain and range of the given function:</h4></ul><div style=margin-left:40px;><h4 class="math question"><em> y </em> = – <em> x </em> <sup> 4 </sup> + 4</h4></div><div class="alignright display-mobile hide-print"style="width:180px; float:right; margin-left:10px;"><p class="in-lesson-resp-ad-header nomargin"align=center>Affiliate<div id=WyzAnt_Widget style=max-width:160px;><div class="wyzant wyzant-widget"data-api-key=db8aced3-1979-4f99-bec1-650e0dfcb678 data-channel-id=162 data-size=3 data-subject=algebra data-widget-type=side-search></div><script data-cookieconsent=marketing src=https://partner-tools.wyzantcdn.com/sprocket/wyzant_widgets.js type=text/plain>
            </script></div></div><div class=indent_prob><p class=text style=padding-top:20px;>This is just a garden-variety polynomial. There are no denominators (so no division-by-zero problems) and no radicals (so no square-root-of-a-negative problems). There are no problems with a polynomial. There are no values that I can't plug in for <span class=math> <em> x </em> </span> . When I have a polynomial, the answer for the domain is <em> always </em> :<div class=indent_prob><p class="text answer">the domain is "all <span class=math> <em> x </em> </span> ".</div><p class=text>The range will vary from polynomial to polynomial, and they probably won't even ask, but when they do, I look at the picture:<p><img alt=graph height=235 src=fcns/domain03.gif width=145><p class=text>The graph goes only as high as <span class=math> <em> y </em> = 4 </span> , but it will go as low as I like. Then:<div class=indent_prob><p class="text answer">The range is "all <span class=math> <em> y </em> ≤ 4 </span> ".</div></div><div class=hide-screen><hr><p id=printout_URL>URL: https://www.purplemath.com/modules/fcns2.htm</div><div class="w3-container hide-print"style="padding:20px 0;"><p style=text-align:center;><a class=w3-btn href=fcns.htm style=background-color:#f1f1f1;> Page 1 </a> <button style="color:#000; background-color:#f1f1f1;"class=w3-btn disabled>Page 2</button></div><div id=mobile_only_box></div><script>window['nitroAds'].createAd('mobile_only_box', {
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