Introduction

What is a graph? Consider first the commonly used concept defined in terms of Cartesian coordinates; figure 1.la shows a graph of a real valued function f of a real continuous variable x where <m:math display=&#x2019;block&#x2019;> <m:semantics> <m:mrow> <m:mi>y</m:mi><m:mo>=</m:mo><m:mi>f</m:mi><m:mo stretchy=&#x2019;false&#x2019;>(</m:mo><m:mi>x</m:mi><m:mo stretchy=&#x2019;false&#x2019;>)</m:mo><m:mo>=</m:mo><m:msup> <m:mi>x</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>&#x2212;</m:mo><m:mn>6</m:mn><m:msup> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo><m:mn>9</m:mn><m:mi>x</m:mi><m:mo>+</m:mo><m:mn>16</m:mn></m:mrow> </m:semantics> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y = f(x) = {x^3} - 6{x^2} + 9x + 16$$<m:math display=&#x2019;block&#x2019;> <m:semantics> <m:mrow> <m:mo>&#x2212;</m:mo><m:mn>1</m:mn><m:mfrac bevelled=&#x2019;true&#x2019;> <m:mn>1</m:mn> <m:mn>4</m:mn> </m:mfrac> <m:mo>&#x2264;</m:mo><m:mi>x</m:mi><m:mo>&#x2264;</m:mo><m:mn>4</m:mn><m:mfrac bevelled=&#x2019;true&#x2019;> <m:mn>3</m:mn> <m:mn>4</m:mn> </m:mfrac> </m:mrow> </m:semantics> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$- 1{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$4$}} \leqslant x \leqslant 4{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 4}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$4$}}$$ Figure 1.1b also shows a graph, this time of <m:math display=&#x2019;block&#x2019;> <m:semantics> <m:mrow> <m:mi>y</m:mi><m:mo>=</m:mo><m:mi>g</m:mi><m:mo stretchy=&#x2019;false&#x2019;>(</m:mo><m:mi>x</m:mi><m:mo stretchy=&#x2019;false&#x2019;>)</m:mo><m:mo>=</m:mo><m:mn>2</m:mn><m:mi>x</m:mi><m:mi>m</m:mi><m:mi>o</m:mi><m:mi>d</m:mi><m:mo stretchy=&#x2019;false&#x2019;>(</m:mo><m:mn>5</m:mn><m:mo stretchy=&#x2019;false&#x2019;>)</m:mo></m:mrow> </m:semantics> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y = g(x) = 2xmod(5)$$<m:math display=&#x2019;block&#x2019;> <m:semantics> <m:mrow> <m:mn>0</m:mn><m:mo>&#x003C;</m:mo><m:mi>x</m:mi><m:mo>&#x003C;</m:mo><m:mn>4</m:mn></m:mrow> </m:semantics> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$0 < x < 4$$ In both cases x can take on a (continuous) infinity of values and the graph consists of continuous lines containing an infinite number of points.