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title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1061/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">2.1 & 2.2 Introducing Functions </span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <div class="intro-card"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/images/2.2-func-pic.png');"> </div> <img src="../../images/2.2-func-pic-1.png" style="display: none"> <div class="content"> <h4 class="theading">What is a function?</h4> <p class="text">Functions are extraordinary algebraic and visual tools which allow us to understand and analyse relationships. In this section you will learn exactly what a function is and how to find a function's inverse, alongside developing a more nuanced sense of important aspects of functions, such as domain and range. </p> </div> </div> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key concepts</p> </div> </div> <div class="panel-body"> <p><span id="docs-internal-guid-00ff9aba-7fff-7710-18c4-d74420756949"></span></p> <p dir="ltr"><span id="docs-internal-guid-00ff9aba-7fff-7710-18c4-d74420756949">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Identify what a function is and know the difference between a one-to-one function and a many-to-one function. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Know the different formats of familiar functions, including linear, quadratic, cubic and exponential functions. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able to identify asymptotes of basic functions. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Find the domain and range of a function. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Understand the connection between a function and its inverse. </p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <p>The following is a series of videos that will help you understand, learn about and review this sub-topic.</p> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Introduction to Functions</p> </div> </div> <div class="panel-body"> <div> <p>What is a function? What's the difference between a one-to-one function, a many-to-one function and a relation? What do we use the vertical line test for?</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/424300632"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Familiar Functions and Function Notation</p> </div> </div> <div class="panel-body"> <div> <p>Can you identify the following functions: linear, quadratic, cubic and exponential? Can you find an output given a input and vice versa?</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/424301170"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> 3. Asymptotes</p> </div> </div> <div class="panel-body"> <div> <p>What is an asymptote and how can you identify where they are on a graph?</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/424302923"></iframe></div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> 4. Domain and Range</p> </div> </div> <div class="panel-body"> <div> <p>What is domain and range? </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/424303278"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Inverse Functions</p> </div> </div> <div class="panel-body"> <div> <p>What is an inverse function?</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/500601568"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div id="carousel-267" class="dynamic-gallery carousel slide" data-id="267"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-267/flashcard1.png" data-fancybox="gallery-267" title="" data-caption=""><img alt="" src="../../../std-galleries/15-267/flashcard1.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-267/flashcard2.png" data-fancybox="gallery-267" title="" data-caption=""><img alt="" src="../../../std-galleries/15-267/flashcard2.png"></a></div></div><a class="left carousel-control" href="#carousel-267" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-267" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-267/flashcard1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-267/flashcard2-thumb128.jpg"><li></li></ol> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <p>Self Checking Quiz</p> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos and above<em>.</em></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-470-2122"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Is this a function or a relation?</p><p><img alt="" src="../../graph-1.png" style="width: 560px; height: 444px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> One-to-one function</label></p><p><label class="radio"><input type="radio"> Many-to-one function</label></p><p><label class="radio"><input class="c" type="radio"> Relation</label></p></div><div class="q-explanation"><p>Since it fails the vertical line test - i.e. there is a section in which a vertical line passes through more than one point on the curve - it must be a relation.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify which functions are linear.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(f(x)=6x+5\)</span></span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(g(x)=10-2x\)</span></span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(w(x)=3x+7+x^2 \)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(q(x)=\frac{1}{3x-2}\)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(v(x)=2^{x+5}\)</span></span></label> </p><div class="ck-mark" id="wrong-mark" style="display:none;"> </div></div><div class="q-explanation"><p>Linear functions - which are straight lines on a co-ordinate grid - have an <span class="math-tex">\(x\)</span> term with a maximum power of <span class="math-tex">\(1\)</span>, which we of course do not write in mathematics. <span class="math-tex">\(w(x)\)</span> is a quadratics function, <span class="math-tex">\(q(x)\)</span> is a rational function and <span class="math-tex">\(v(x)\)</span> is an exponential function.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)=\frac{1}{3}x^2+\frac{2}{5}x-2\)</span></p><p>Find <span class="math-tex">\(f(2)\)</span>, giving your answer to 3 significant figures. </p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.133"> <span class="review"></span> </p></div><div class="q-explanation"><p>If you substitute <span class="math-tex">\(x=2\)</span> into the functin you obtain 0.133.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A toy sales person sells <span class="math-tex">\(t\)</span> toys per month. </p><p>Her monthly profit in US dollars can be modelled by: <span class="math-tex">\(N(t)=t^3+2t-100\)</span></p><p>Find the value of <span class="math-tex">\(N\)</span> if no toys are sold.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-100"> <span class="review"></span></p></div><div class="q-explanation"><p>If no toys are sold, <span class="math-tex">\(t=0\)</span>, and hence <span class="math-tex">\(N(0)=-100\)</span> dollars. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)=\frac{3+x}{x^2}\)</span></p><p>A point <span class="math-tex">\((x, 14)\)</span> lies on this function. Find the value of <span class="math-tex">\(x \)</span></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.25"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by substituting <span class="math-tex">\(14\)</span> into the equation: <span class="math-tex">\(14=\frac{3+x}{x^2}\)</span></p><p>Using your GDC, graph each side of the equation and find the point of interestion of these graphs.</p><p>Since we are only interested in the <span class="math-tex">\(x\space co-ordinate\)</span> of that intersection point, you find <span class="math-tex">\(x=0.25\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>If you know that <span class="math-tex">\(f(10)=25\)</span>, find <span class="math-tex">\(f^{-1}(25)\)</span></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="10"> <span class="review"></span> </p></div><div class="q-explanation"><p>If <span class="math-tex">\(f\)</span> takes an input of <span class="math-tex">\(10\)</span> and produces an output of <span class="math-tex">\(25\)</span>, then the inverse of <span class="math-tex">\(f\)</span>, <span class="math-tex">\(f^{-1}\)</span>, does the opposite. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the domain of the function, <span class="math-tex">\(f(x)=\sqrt{x}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(x \geq 0\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x > 0\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x \geq 1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x \leq 0\)</span></span></label> </p></div><div class="q-explanation"><p>If you input the graph of <span class="math-tex">\(f(x)=\sqrt{x}\)</span> into your GDC, you will notice that it does not exist for any negative value of <span class="math-tex">\(x \)</span>, because we cannot take the square root of a negative number and obtain a real answer. The domain is therefore, <span class="math-tex">\(x \geq 0\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the range of the function <span class="math-tex">\(f(x)=x^2\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(f(x)>0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(f(x) <0\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(f(x) \geq 0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(f(x) \geq1\)</span></label></p></div><div class="q-explanation"><p>If you completed the previous question, then you will already be aware the the domain of a function, is the same the range of its inverse. Therefore, the range of <span class="math-tex">\(f(x)=x^2 \)</span> is <span class="math-tex">\(f(x) \geq0\)</span>. This can also been seen through inputting the graoh into your GDC, or noting that when you square any input number, the output will always be positive or zero. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>What can you say about the domain and range of any diagonal straight line (linear function).</p></div><div class="q-answer"><p>Domain: <input type="text" style="height: auto;" data-c="all real numbers"> <span class="review"></span> </p><p>Range: <input type="text" style="height: auto;" data-c="all real numbers"> <span class="review"></span> </p><p>Choose options below to write into the text input boxes - note that it may be the same option. </p><p>a. all real numbers</p><p>b. x / f(x) must be greater than zero</p><p>c. x / f(x) must be less than zero</p></div><div class="q-explanation"><p>The domain and range of diagonal straight lines is all real numbers. This is not the case for horizontal or vertical lines. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>To graphically obtain an inverse function, you reflect it on the co-ordinate grid in the diagonal line, .....................................</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>y = x</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = 0</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = -x</span></label> </p><p><label class="radio"> <input type="radio"> <span>x = 0</span></label> </p></div><div class="q-explanation"><p>To obtain the inverse function graphically, reflect it in the diagonal line, y = x. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam Style Questions</p> </div> </div> <div class="panel-body"> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../images/exam-q1.png" style="width: 600px; height: 387px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/457903610"></iframe></div> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../images/picture1(8).png" style="width: 600px; height: 182px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/457904013"></iframe></div> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../images/picture2(1).png" style="width: 600px; height: 195px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/457904258"></iframe></div> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <h4> </h4> <div class="page-container panel-self-assessment" data-id="2122"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>2.1 & 2.2 Introducing Functions </strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label 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