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      <text>2.3.1 (Functions and relations)</text>
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      <text><![CDATA[<h1>Introduction to Functions and Relations</h1>

<p>Functions and relations are one of the most important topics in mathematics. The idea of a mathematical function allows us to name the process of of evaluating a particular expression so that we can discuss it as a whole. Functions act like a machine where we can give it an input and we receive an output according to the function rule. This is why almost everything in real life that we model mathematically is modelled through a type of function, because it is a way of representing a relationship between variables, e.g. if we wanted to examine how speed of a marathon runner is affected by time. This is the beginning of calculus and therefore the beginning of understanding the physical world. It transpires that the notion of a function is highly adaptable, and if you do more maths in your studies, you will see how we can write functions of several variables and several dimensions. However in this section we will focus on the notion of functions explicitly and examine the kind of manipulations we can perform with functions. So far in the course, we have met several examples of functions, for example quadratics and polynomials and we have seen how it is often useful to examine these functions via their graphs. We shall expand our inventory of common functions as we progress through the course, but for now we shall discuss the basics of function theory.</p>
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<p>Firstly we need to define a relation. Note that a relation and a function are different but related concepts. A <strong>relation</strong> is <em>any</em> set of points which connect two variables. Most often, we represent a relation as an equation, usually with the variables \(x\) and \(y\). For example \(y = x^2\) is an equation which represents a relation. We should note however that not every relation can be defined by an equation.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Relations%20Comparison.png" alt="Comparison of relations with equations and without" width="651" height="300" /></p>
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<p>A <strong>function</strong> is a relation in which no two different ordered pairs have the same first component. In other words a function is a relation where each input has exactly one output. However, note that an output can be obtained from two different inputs as long as the output is the only output for each input. For example \(y=x^2\) is a function because whilst \(x = \pm 2\) both give \(y =4\), this is the only possible value for both \(x=-2\) and \(x=2\). In terms of graphs this means that for each value of \(x\) there is one value of \(y\). Therefore a function is a special type of relation, every function is a relation but not every relation is a function.</p>
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<p style="text-align: center;"><img src="@@PLUGINFILE@@/Function%20vs%20not%20function.png?time=1598615075915" alt="Diagram of a function vs not a function" width="780" height="348" /></p>]]></text>
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