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<title>Domain, Range and Codomain</title>
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<meta name="description" content="Learn about the differences between Domain, Range and Codomain. In its simplest form the domain is all the values that go into a function ... " />
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<h1 align="center">Domain, Range and Codomain</h1>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/range-domain-graph.svg" alt="doman and range graph" /></p>
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<p>In its simplest form the domain is all the values that go into a function, and the range is all the values that come out.</p>
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<p>But in fact they are very important in <b>defining</b> a function. Read on!</p>
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<p align="center"><i>Please read "<a href="function.html">What is a Function?</a>" first ...</i></p>
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<h2>Functions</h2>
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<p class="larger">A function <i><b>relates</b></i> an input to an output:</p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/tree.jpg" width="172" height="148" alt="tree" /></p>
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<p>Example: this tree grows 20 cm every year, so the height of the tree is <i><b>related</b></i> to its age using the function <i><b>h</b></i>:</p>
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<p align="center"><b><i>h</i>(age) = age × 20</b></p>
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<p>So, if the age is 10 years, the height is <i>h</i>(10) = 200 cm</p>
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<p class="larger">Saying "<b><i>h</i>(10) = 200</b>" is like saying 10 is related to 200. Or 10 → 200</p>
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</div>
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<h2>Input and Output</h2>
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<p>But not all values may work! </p>
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<ul>
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<li>The function may not work if we give it the wrong values (such as a negative age), </li>
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<li>And knowing the values that can come out (such as always positive) can also help</li>
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</ul>
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<p>So we need to say all the values that <b>can go into</b> and <b>come out of</b> a function. </p>
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<p>This is best done using<b> <a href="sets-introduction.html">Sets</a> ...</b></p>
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<div class="simple">
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<table width="86%" border="0" align="center">
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<tr>
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<td width="17%"><img src="images/various-reals.svg" alt="various real numbers" /> </td>
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<td width="83%" valign="middle">
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<h3>A set is a collection of things, such as numbers. </h3>
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<p>Here are some examples:</p>
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<div class="indent50px">Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
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<br /> Set of odd numbers: {..., -3, -1, 1, 3, ...}
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<br /> Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
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<br /> Positive multiples of 3 that are less than 10: {3, 6, 9}</div>
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</td>
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</tr>
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</table>
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</div>
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<p>In fact, a function is defined in terms of sets:</p>
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<table border="0" align="center">
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<tr>
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<td align="center">
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<h2>Formal Definition of a Function</h2>
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<h3>A function relates each element of a set<br />
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with exactly one element of another
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set<br />
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(possibly the same set).</h3> </td>
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<td align="center"><img src="images/function-sets.svg" alt="function sets X to Y" /></td>
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</tr>
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</table>
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<h2>Domain, Codomain and Range</h2>
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<p>There are special names for <b>what can go into</b>, and <b>what can come out</b> of a function:</p>
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<table width="92%" border="0" align="center">
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<tr>
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<td width="60"><img src="../images/style/yes.svg" height="46" class="in" alt="yes" /></td>
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<td>What can go <b>into</b> a function is called the <b>Domain</b></td>
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</tr>
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<tr>
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<td width="60"><img src="../images/style/yes.svg" alt="yes" height="46" class="in" /></td>
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<td>What <b>may possibly come out</b> of a function is called the <b>Codomain</b></td>
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</tr>
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<tr>
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<td width="60"><img src="../images/style/yes.svg" alt="yes" height="46" class="in" /></td>
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<td>What <b>actually comes out</b> of a function is called the <b>Range</b></td>
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</tr>
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</table>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/domain-range-codomain.svg" alt="Domain, Range and Codomain for x to 2x+1" /></p>
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<h3>Example </h3>
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<p>• The set "A" is the <b>Domain</b>, </p>
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<p>• The set "B" is the <b>Codomain</b>, </p>
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<p>• And the set of elements that get pointed to in B (the actual values produced by the function) are the <b>Range</b>, also called the Image. </p>
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<p>And we have:</p>
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<ul>
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<li>Domain: {1, 2, 3, 4}</li>
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<li>Codomain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}</li>
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<li>Range: {3, 5, 7, 9}</li>
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</ul>
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</div>
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<h2>Part of the Function</h2>
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<p>Now, what comes <b>out</b> <i>(the Range)</i> depends on what we put <b>in</b> <i>(the Domain)</i> ...</p>
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<p align="center" class="larger"> ... but <b>WE</b> can define the Domain!</p>
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<p>In fact the Domain is an essential part of the function. Change the Domain and we have a different function.</p>
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<div class="example">
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<p>Example: a simple function like f(x) = x<sup>2</sup> can have the <b>domain</b> (what goes in) of just the counting numbers {1,2,3,...}, and the <b>range</b> will then be the set {1,4,9,...}</p>
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<p align="center"><img src="images/range-domain.svg" alt="Domain to Range f(x) = x^2" /></p>
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<p>And another function g(x) = x<sup>2</sup> can have the domain of integers {...,-3,-2,-1,0,1,2,3,...}, in which case the range is the set {0,1,4,9,...}</p>
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<p align="center"><img src="images/range-domain2.svg" alt="Domain to Range g(x) = x^2" /></p>
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</div>
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<table width="90%" border="0" align="center">
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<tr>
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<td width="10%"><img src="../images/style/run.jpg" width="50" height="50" class="in" alt="run" /></td>
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<td width="90%">
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<p>Even though both functions take the input and square it, they have a <b>different set of inputs</b>, and so give a different set of outputs. </p>
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<p>In this case the range of g(x) also includes 0.</p>
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</td>
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</tr>
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<tr>
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<td width="10%"> </td>
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<td width="90%"> </td>
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</tr>
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<tr>
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<td width="10%"><img src="../images/style/pencil-paper.gif" width="39" height="35" alt="pencil paper" /></td>
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<td width="90%">
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<p>Also they will have different properties.</p>
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<p>For example f(x) always gives a unique answer, but g(x) can give the same answer with two different inputs (such as <b>g(-2)=4</b>, and also <b>g(2)=4</b>)</p>
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</td>
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</tr>
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</table>
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<p>So, the domain is an essential part of the function. </p>
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<h2>Does Every Function Have a Domain?</h2>
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<p>Yes, but in simpler mathematics we never notice this, because the domain is <b><i>assumed</i></b>:</p>
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<ul>
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<li>Usually it is assumed to be something like "all numbers that will work". </li>
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<li>Or if we are studying whole numbers, the domain is assumed to be whole numbers. </li>
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<li>etc.</li>
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</ul>
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<p>But in more advanced work we need to be more careful!</p>
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<h2>Codomain vs Range</h2>
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<p>The Codomain and Range are both on the output side, but are subtly different.</p>
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<p>The Codomain is the set of values that could <b>possibly</b> come out. The Codomain is actually <b>part of the definition</b> of the function. </p>
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<p>And The Range is the set of values that <b>actually do</b> come out.</p>
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<div class="example">
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<p>Example: we can define a function <i><b>f(x)=2x</b></i> with a domain and codomain of integers (because we say so). </p>
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<p>But by thinking about it we can see that the range (actual output values) is just the <b>even</b> integers.</p>
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<p>So the codomain is integers (we defined it that way), but the range is even integers.</p>
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</div>
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<p>The Range is a subset of the Codomain.</p>
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<p><b>Why both?</b> Well, sometimes we don't know the <i><b>exact</b></i> range (because the function may be complicated or not fully known), but we know the set it <i><b>lies in</b></i> (such as integers or reals). So we define the codomain and continue on.</p>
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<h2>The Importance of Codomain</h2>
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<p>Let me ask you a question: Is <b><i><a href="../algebra/square-root.html">square root</a></i> </b> a function?</p>
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<p>If we say the codomain (the possible outputs) is <b>the set of real numbers</b>, then square root is <b>not a function</b>! ... is that a surprise? </p>
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<p>The reason is that there could be two answers for one input, for example <b><i>f(9) = 3 </i></b>or<b><i> -3</i></b></p>
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<div class="def">
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<p>A <a href="function.html">function</a> must be <i><b>single valued</b></i>. It cannot give back 2 or more results for the same input. So "f(9) = 3 <b>or</b> -3" is not right! </p>
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</div>
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<p>But it can be fixed by simply <b>limiting the codomain</b> to non-negative real numbers.</p>
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<div class="def">
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<p><span class="huge">√</span>In fact, the radical symbol (like <b>√x</b>) always means the principal (positive) square root, so <b>√x</b> is a function because its codomain is correct.</p>
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</div>
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<p>So, <b>what we choose for the codomain </b>can actually affect whether something is a <b>function or not.</b> </p>
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<p> </p>
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<h2>Notation</h2>
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<p>Mathematicians don't like writing lots of words when a few symbols will do. So there are ways of saying "the domain is", "the codomain is", etc.</p>
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<p>This is the neatest way I know:</p>
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<table width="90%" border="0" align="center">
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<tr>
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<td width="29%">
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<div align="center"><img src="images/f-n-n.png" width="84" height="20" alt="f : N to N" /></div>
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</td>
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<td width="71%">
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<p>this says that the function "<i><b>f</b></i>" has a domain of "<b>N</b>" (the <a href="../whole-numbers.html">natural numbers</a>), and a codomain of "<b>N</b>" also.</p>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td width="29%">
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<div align="center"><img src="images/f-x-x2.png" width="97" height="25" alt="f : x to x^2" />
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<br /> <i>or</i>
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<br /> <img src="images/f-x-x2b.png" alt="f(x)=x^2" width="82" height="23" /> </div>
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</td>
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<td width="71%">
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<p>and either of these say that the function "f" takes in "x" and returns "x<sup>2</sup>"</p>
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</td>
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</tr>
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</table>
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<p>There is also:</p>
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<p align="center"><span class="larger">Dom(f)</span> or <span class="larger">Dom f</span> meaning "the domain of the function f"</p>
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<p align="center"><span class="larger">Ran(f)</span> or <span class="larger">Ran f</span> meaning "the range of the function f"</p>
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<h2>How to Specify Domains and Ranges</h2>
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<p>Learn how to specify Domains and Ranges at <a href="set-builder-notation.html">Set Builder Notation</a>.</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">
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getQ(536, 537, 538, 539, 1174, 1175, 1176, 2341, 2342, 2343);
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</script> </div>
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<div class="related"> <a href="function.html">What is a Function?</a> <a href="injective-surjective-bijective.html">Injective, Surjective and Bijective</a> <a href="sets-introduction.html">Introduction to Sets</a> <a href="index.html">Sets Index</a> </div>
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Copyright © 2018 MathsIsFun.com
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