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fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../539/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Functions - The Basics</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"> <p><img alt="" src="../../files/functions/the-basics/main2-1.png" style="float: left; width: 100px; height: 100px;"> In this page, we will look at the different types of relations including functions. The first step to gaining a good understanding of functions is to think about inputs and outputs. Once that is clear, you should be able to deal with domain and range, as well as composite and inverse functions. For the IB examination, questions require you to have a strong conceptual understanding of functions, with questions often posed with graphs. You will be well prepared for that if you work through this page.</p> <hr class="hidden-separator"> <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"> <div> <p>On this page, you should learn to</p> <ul> <li>Understand the different types of relation: one-to-one function, many-to-one- function, one-to-many relation, many-to-many relation</li> <li>Use the domain and range of a function</li> <li>Find and use inverse functions <span class="math-tex">\(f^{-1}\)</span></li> <li>Use composite functions <span class="math-tex">\((f∘g)(x)=f(g(x))\)</span></li> <li>"Draw" and "Sketch" a graph</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/functions/the-basics/revision-notes_functions_basics.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/the-basics/revision-notes_functions_basics.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that tests your understanding of the basics of functions, the different types of relation, and also the domain and range of functions:</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#adafe138"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="adafe138"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Functions - the basics <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-457-676" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>f(x) = 2x² - 1</p><p>Work out <strong>f(-1)</strong></p></div><div class="q-answer"><p><strong>f(-1)</strong> =<input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>f(-1) = 2(-1)² - 1</p><p>f(-1) = 2(1) - 1=1</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>f(x) = 2<sup>x</sup> + 1</p><p>Work out <strong>f(3)</strong></p></div><div class="q-answer"><p><strong>f(3) </strong>= <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>f(3) = 2<sup>3</sup> + 1</p><p>f(3) = 8 + 1 = 9</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(f(x)=\frac{2}{x-1}\)</span></p><p><strong><span class="math-tex">\(f(a)=0.5\)</span></strong></p><p>Work out <strong>a</strong></p></div><div class="q-answer"><p><strong>a</strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{2}{a-1}=0.5\)</span></p><p>2 = 0.5a - 0.5</p><p>2.5 = 0.5a</p><p>a = 5</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Which of the following best describes this relation?</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q4.png" style="width: 300px; height: 308px;"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> one-to-one function</label></p><p><label class="radio"><input type="radio"> many-to-many relation</label></p><p><label class="radio"><input type="radio"> many-to-one function</label></p><p><label class="radio"><input type="radio"> one-to-many relation</label></p></div><div class="q-explanation"><p>Each input is mapped on to one and only one output</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Which of the following best describes this relation?</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q5.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> one-to-many relation</label></p><p><label class="radio"><input type="radio"> one-to-one function</label></p><p><label class="radio"><input type="radio"> many-to-many relation</label></p><p><label class="radio"><input class="c" type="radio"> many-to-one function</label></p></div><div class="q-explanation"><p>Each possible output (except y = 1) has two corresponding inputs</p><p>E.g. 1→2, 3→2</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q5ans.bmp" style="width: 400px; height: 258px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Which of the following best describes this relation?</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q6.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> one-to-one function</label></p><p><label class="radio"><input class="c" type="radio"> many-to-many relation</label></p><p><label class="radio"><input type="radio"> one-to-many relation</label></p><p><label class="radio"><input type="radio"> many-to-one function</label></p></div><div class="q-explanation"><p>Each output has two inputs</p><p>Each input has two outputs</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q6ans.bmp" style="width: 400px; height: 258px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Which of the following is the domain of this function</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q7.bmp" style="width: 400px; height: 258px;"></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 type="radio"> <span class="math-tex">\(x>0\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(x\in \mathbb{R}\)</span></label></p></div><div class="q-explanation"><p>The domain is all the possible inputs.</p><p>The inputs of this function are the set of real numbers (all real numbers are possible)</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Which of the following is the range of this function</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q8.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x \in \mathbb{R}, x\neq1\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(f(x) \in \mathbb{R}, f(x) \neq0\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x>1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f(x) \in \mathbb{R}\)</span></span></label> </p></div><div class="q-explanation"><p>The range is all the possible outputs.</p><p>The outputs of this function are the set of real numbers except f(x) = 0 (all real numbers are possible except 0)</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The range of the following function is a ≤ f(x) ≤ b</p><p>Find the values of <strong>a</strong> and <strong>b</strong></p><p><img alt="" src="../../files/functions/the-basics/quizzes/q9.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><strong>a</strong> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p><strong>b</strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p>The range is all the possible outputs.</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q9ans.bmp" style="width: 400px; height: 258px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Let <span class="math-tex">\(f(x)=1-(x-2)^2\)</span>, x ≥ <strong>k</strong> is a one-to-one function</p><p>Find the smallest value of <strong><em>k</em></strong></p></div><div class="q-answer"><p><strong>k</strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The graph of the function with an unrestricted domain looks like this</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q10.bmp" style="width: 400px; height: 258px;"></p><p>For it to be a one-to-one function, we have to restrict the domain like this</p><p><img alt="" src="../../files/functions/the-basics/quizzes/q10ans.bmp" style="width: 400px; height: 258px;"></p><p>...we can find the inverse of this function now</p></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz that tests your understanding of composite and inverse functions</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#04c28988"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="04c28988"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Functions - composite and inverse <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-458-676" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Let <span class="math-tex">\(f(x)=\frac{x^3}{2}\)</span></p><p>Work out <span class="math-tex">\(f^{-1}(32)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(f^{-1}(32)\)</span> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>You do not have to work out the inverse function.</p><p>What is the input of the function <strong><em>f</em></strong> that gives an ouptut of 32 ?</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q1ans.png" style="width: 200px; height: 195px;"></p><p><span class="math-tex">\(\frac{x^3}{2}=32\)</span></p><p>x<sup>3 </sup>= 64</p><p>x = 4</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Given that</p><p>f(x) = x + 1</p><p>g(x) = x² - 1</p><p>h(x) = 0.5x</p><p>Work out fgh(4)</p></div><div class="q-answer"><p>fgh(4) = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>There is no need to work out the composite function f(g(h(x)))</p><p>Make sure you do the functions in the correct order: 'do' h, then 'do' g, then 'do' f</p><p>h(4) = 0.5x4 = 2</p><p>g(2) = 2² - 1 = 3</p><p>f(3) = 3 + 1 = 4</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is plotted.</p><p>The domain of <strong>f<sup>-1</sup></strong> is <strong>a ≤ x ≤ b</strong></p><p>Work out <strong>a</strong> and <strong>b</strong></p><p><strong><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q3.bmp" style="width: 400px; height: 258px;"></strong></p></div><div class="q-answer"><p><strong>a</strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p><p><strong>b</strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>The domain of<strong> f<sup>-1</sup></strong> is the range of <strong>f</strong></p><p>This is all the possible outputs of the function<strong> f </strong>from the graph above (all possible y values)</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is plotted.</p>Use it to work out<strong> f<sup>-1</sup>(1)</strong><p><strong><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q3.bmp" style="width: 400px; height: 258px;"></strong></p></div><div class="q-answer"><p><strong>f<sup>-1</sup>(1) </strong>= <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>The output of <strong>f</strong> is 1. What is the input?</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q3ans.bmp" style="width: 400px; height: 257px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graph of the function y = f(x) is plotted</p><p>Use it to work out <strong>f<sup>-1</sup>(0)</strong></p><p><strong><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q5.bmp" style="width: 400px; height: 258px;"></strong></p></div><div class="q-answer"><p><strong>f<sup>-1</sup>(0)</strong> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>The output of <strong>f</strong> is 0. What is the input?</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q5ans.bmp" style="width: 400px; height: 258px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Let f(x) = 2x - 1</p><p>The graph of y = g(x) is shown</p><p>Work out <strong>fg(1)</strong></p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q6.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><strong>fg(1)</strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>We 'do' g first</p><p>From the graph g(1) = 2</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q6ans.bmp" style="width: 400px; height: 258px;"></p><p>Then we 'do' f</p><p>f(2) = 2x2 - 1 = 3</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Let f(x) = 2<sup>x</sup></p><p>The graph of y = g(x) is shown</p><p>Work out <strong>gf(2)</strong></p><p><strong><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q7.bmp" style="width: 400px; height: 258px;"></strong></p></div><div class="q-answer"><p><strong>gf(2) </strong>= <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>We 'do' f first</p><p>f(2) = 2<sup>2</sup> = 4</p><p>Then, we 'do' g.</p><p>From the graph g(4) = 5</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q7ans.bmp" style="width: 400px; height: 258px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graphs of y = f(x) and y = g(x) are plotted</p><p>Work out <strong>fg(2)</strong></p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q8.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><strong>gf(2)</strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p>We 'do' g first:</p><p>From the graph g(2) = 3</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q8ans1.bmp"></p><p>Now we 'do' f:</p><p>From the graph f(3) = 7</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q8ans2.bmp" style="width: 400px; height: 258px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is plotted</p><p>Let <span class="math-tex">\(g(x)=\sqrt{x+3} \ , \ x \neq-3\)</span></p><p>Work out <strong>g(f<sup>-1</sup>(-2))</strong></p><p><strong><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q9.bmp" style="width: 400px; height: 258px;"></strong></p></div><div class="q-answer"><p><strong>g(f<sup>-1</sup>(-2))</strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>From the graph, <strong>f<sup>-1</sup>(-2) </strong>= 6</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q9ans.bmp" style="width: 400px; height: 258px;"></p><p><span class="math-tex">\(g(6)=\sqrt{6+3} =3\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The graph of y = f(x) is plotted</p><p>Work out <strong>f<sup>-1</sup>[f<sup>-1</sup>(4)]</strong></p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q10.bmp" style="width: 400px; height: 258px;"></p></div><div class="q-answer"><p><strong>f<sup>-1</sup>[f<sup>-1</sup>(4)]</strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>From the graph f<sup>-1</sup>(4) = 1</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q10ans1.bmp" style="width: 400px; height: 258px;"></p><p>From the graph f<sup>-1</sup>(1) = -1</p><p><img alt="" src="../../files/functions/the-basics/quizzes/compoundandinverse/q10ans2.png" style="width: 400px; height: 250px;"></p></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </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"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <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"> <div> <div class="smart-object center" data-id="864"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The function <strong><em>f</em></strong> and<strong><em> g</em></strong> are defined as follows</p> <p><span class="math-tex">\(f(x)=3x^2 \ ,\ x\in\mathbb{R}\)</span></p> <p><span class="math-tex">\(g\left(x\right)=\frac{2}{x}-4\ ,x\in\mathbb{R},\ x>0\)</span></p> <p>a) Find the range of <strong><em>g</em></strong></p> <p>b) Find g<sup>-1</sup></p> <p>c) Solve gf(x) = 2</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) To find the range, consider the graph of the function. What are all the possible y values?</p> <p><content>c) Careful x > 0</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics1a.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border 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"> <div> <div class="smart-object center" data-id="863"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <strong><em>f </em></strong>and <strong><em>g</em></strong> be the functionsLet <strong><em>f </em></strong>and <strong><em>g</em></strong> be the functions</p> <p><span class="math-tex">\(f(x)=ln(1-2x),x<\frac{1}{2}\)</span></p> <p><span class="math-tex">\(g(x)=\frac{4}{x-2},x\neq2\)</span></p> <p>a) Find the exact value of fg(-2)</p> <p>b) Find f<sup>-1</sup>(x), stating its domain</p> <p>c) Show that <span class="math-tex">\(gg(x)=\frac{4x-8}{8-2x}\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>b) domain of f<sup>-1 </sup>is the range of f</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border 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"> <div> <div class="smart-object center" data-id="865"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The graph of the function <strong><em>f</em></strong> is plotted below</p> </div> <p>Use the graph to find values of</p> <p>a) <span class="math-tex">\(ff(-2)\)</span></p> <p>b) <span class="math-tex">\(f^{-1}(0)\)</span></p> <p>Let <strong><em>g</em></strong> be the function such that g(x) = 4x + 2</p> <p>c) Work out fg(1)</p> <p>d) Draw a sketch of <span class="math-tex">\(y=f^{-1}(x)\)</span></p> <p><img alt="" src="../../files/functions/the-basics/esq3" style="width: 500px; height: 397px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Think of this question in terms of inputs and outputs</p> <p><content>f(-2) is the output from the function f when we input 2. From the graph, what is the y value, if the x value is -2?</content></p> <p>f(-2) = 1</p> <p>ff(-2) is the value that we get when the output from f(-2) becomes the new input into the function f</p> <p>ff(-2) = 2</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="866"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=\frac{ax-1}{x-a},\ x\neq\ a\)</span></p> <p>a) Show that <span class="math-tex">\(f^{-1}(x)=f(x)\)</span></p> <p>b) Hence, find ff(b)</p> <p>c) The graph of y = g(x) is shown below. Find the values of <strong><em>a</em></strong> if fg(3) = a + 5</p> <p><img alt="" src="../../files/functions/the-basics/esq4" style="width: 500px; height: 385px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">b) Notice the word 'hence'. This means that you should use what you found out in part a to answer this question<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="893"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let the function <strong><em>f</em></strong> be defined by</p> <p><span class="math-tex">\(f(x)=\frac{x-k}{2-x},x\neq2\)</span></p> <p>a) Find <span class="math-tex">\(f^{-1}(x)\)</span> in terms of <strong><em>k</em></strong></p> <p>b) Find <strong><em>k</em></strong> if <span class="math-tex">\(y=f(x)\)</span> and <span class="math-tex">\(y=f^{-1}(x)\)</span> intersect at (-2 , -2) and (3 , 3)</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Solve <span class="math-tex">\(f(x)=f^{-1}(x)\)</span><content></content></p> <p><span class="math-tex">\(\frac{x-k}{2-x}=\frac{2x+k}{1+x}\)</span></p> </section> <h4>Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Print from <a href="../../files/functions/the-basics/esq_functions_basics5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="676"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label 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