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<mdoxml version="1.0"><p>You work in a maths shop in the lobby of the <a href="https://nrich.maths.org/5788">Hilbert Hotel</a>.</p>
<p> </p>
<p>The manager wants to make a window display to highlight the different types of real valued functions of the real numbers that she has on offer, with the 'best' examples of functions from each category along with a representation which most sums up the 'essence' of each category. She decides that she wants to showcase 9 particularly important types of function categories:</p>
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<p>1 Periodic</p>
<p>2 Tending to a vertical asymptote</p>
<p>3 Discontinuous somewhere</p>
<p>4 Decreasing</p>
<p>5 Bounded</p>
<p>6 Infinitely differentiable at all points</p>
<p>7 Singular somewhere</p>
<p>8 Taking finitely many values</p>
<p>9 Unique tangent exists at all points</p>
<p> </p>
<p>Think of a few examples of functions from each category and the different ways that you might represent the different categories. What would be the clearest examples and representations that you could think of to showcase these function categories?</p>
<p> </p>
<p>It might be that you are in competition with another assistant to produce the best display; if so you will need to convince the manager that your selection of 9 functions and representations is the best; it may be that you will need to work collaboratively simply to dream up any examples in some of the categories! It might be that you wish to suggest a better set of function categories.</p>
<p> </p>
<p>Imagine now that you are faced with fussy customers who are likely to request simple examples of functions satisyfing <em>pairs</em> of these properties. Which requests can you satisfy? Which requests will it be impossible to satisfy?</p>
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<h3>Why do this problem?</h3>
<p><a href="https://nrich.maths.org/6904">This problem</a> introduces students to the concept of different categories of real functions which permeate advanced mathematics. It focuses on understanding the properties of the categories as a whole rather than the properties of individual examples. Hopefully students will leave with the realisation that smooth functions are a very special group of
functions along with a wider understanding of functions, continuity and differentiability.</p>
<h3> </h3>
<h3>Possible approach</h3>
<div>This problem assumes that students will have encountered informally ideas of continuity and differentiability. You can role play this task, or play it straight as you wish.</div>
<div> </div>
<div>Start by asking the students (individually or in groups) to come up with a single example of a function from each category, and then share these collectively. Key issues which are likely to occur here are 1) Some functions are members of multiple categories and 2) students will probably want to sketch examples for some categories.</div>
<div> </div>
<div>Once the students are involved with the concept, ask them to produce the 'best' example of a function from each category along with a representation of the category as a whole, as if for a small display poster which would sit in a shop window.</div>
<div> </div>
<div>Note: possible representations of the categories are: multiple algebraic examples, sketches/graphs, descriptions in words (such as 'functions which can be drawn without taking the pen off the paper') or formal mathematical descriptions, such as $f(x+Na) = f(x)$.</div>
<div> </div>
<div>Once students have worked on their ideas they can share them, where the challenge is for others to guess the category. A 'good' solution to this part will be one which clearly points to one and only one category. Be prepared for multiple suggestions and representations and you might find differences of opinion concerning which solutions are 'best'. You can put some of these on the walls of
your classroom to remind students of the meaning of the different function categories.</div>
<div> </div>
<div>The last part of the question is more traditionally algebraic and can be attempted independently if you wish. Simply tabulate a grid with each property as a row and column and ask students to find algebraic examples of functions which fit in the different grid cells. How many grid cells is it possible in principle to fill? Are members of any categories automatically members of other
categories? Do any particularly interesting examples crop up?</div>
<div> </div>
<div> </div>
<h3>Key questions</h3>
<p>Do you understand all of the terms?</p>
<p>In what ways can you describe or represent a function?</p>
<p>How would you describe in words the 'essence' of each function category?</p>
<div> </div>
<h3>Possible extension</h3>
<div>(Difficult) Ask students to find examples of functions which fit none of these categories. What other sorts of categories of functions would be needed to accommodate their new examples?</div>
<div> </div>
<h3>Possible support</h3>
<div>Focus on curve sketches which best represent each category.</div>
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<mdoxml version="1.0"><p>Recall that a function is any rule which assigns a unique value in the range of the function to any value in the domain of the function. It is common to encounter functions which can be expressed through simple algebraic equations, but this is not the only way to define functions.</p>
<p> </p>
<p>To get you started you might think about these sorts of functions: polynomials, trig functions, exponentials, integer powers of the variable and decide in which function categories these belong.</p>
<p> </p>
<p>You might then consider some more complicated functions. For example, some might be defined in two parts such as $f(x) = A(x), x\geq 0$ and $f(x) = B(x), x&lt;0$.</p>
<p> </p>
<p>Don't be afraid to be inventive!</p></mdoxml>
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Maths Shop Window
Make a functional window display which will both satisfy the manager and make sense to the shoppers
Using, Applying and Reasoning about Mathematics
Representing
Sequences, Functions and Graphs
Functions and their inverses
Pre-Calculus and Calculus
Calculus generally