<text><span class="style10">unctions, Graphs and Change (2 of 6)</span><span class="style7">Here the independent and the dependent variables of a function are represented by two lines at right angles (the </span><span class="style26">x</span><span class="style7">-axis and the </span><span class="style26">y</span><span class="style7">-axis) that cross at the origin. The curve representing the function is then the line that passes through the points whose coordinates satisfy the function. For example, the curve of the function </span><span class="style26">y</span><span class="style7">= </span><span class="style26">x</span><span class="style7"> to the power of 2 is the set of pairs, (</span><span class="style26">x, y</span><span class="style7">), of real numbers for which </span><span class="style26">y</span><span class="style7"> is the square of </span><span class="style26">x</span><span class="style7">; thus, for example, (2,4), (-1,1), (-2,4), (2,2), etc., are all in the graph of the function. The curve corresponding to this function is shown below. This system of coordinates is named after the French philosopher and mathematician René Descartes (or des Cartes, whence the adjective 'Cartesian'.</span><span class="style10">Graphs and curves</span><span class="style7">Because a function associates elements of one set with those of another, it defines the set of all pairs of elements, (</span><span class="style26">x,y</span><span class="style7">), in which </span><span class="style26">x</span><span class="style7"> is a value of the independent variable and </span><span class="style26">y</span><span class="style7"> is the value of the function for the argument </span><span class="style26">x</span><span class="style7">. Another way of expressing this is that any point that </span><span class="style26">satisfies</span><span class="style7"> the function </span><span class="style26">y = f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">) can be represented by the point (</span><span class="style26">x, f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">)). Since a function must be a many-one relation, every such pair has a different first element, so the pairs can be listed in a unique order. The function can then be thought of as moving through the values of the dependent variable as the value of the independent variable increases. This is what is represented by a graph in the Cartesian coordinate system: if we now draw a line joining the points (</span><span class="style26">x, f</span><span class="style7">(</span><span class="style26">x</span><span class="style7">)) as </span><span class="style26">x</span><span class="style7"> increases, this line passes through all and only the points whose coordinates satisfy the function. Such a line is usually called a </span><span class="style26">graph</span><span class="style7">, although mathematicians prefer to use that term for the set of values of the variables, and call the diagram a </span><span class="style26">curve</span><span class="style7">. Since this way of representing change and dependency is equivalent to the function itself, curves provide us with a way of visualizing processes of change.</span></text>
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<text>ΓÇó MOTION AND FORCEΓÇó MATHEMATICS AND ITS APPLICATIONSΓÇó SETS AND PARADOXESΓÇó CORRESPONDENCE, COUNTING AND INFINITY</text>
