1Functions and their graphsActivity 12Sketching the graphs of functions whose domains arenot the largest sets of numbers for which their rules are applicableSketch the graphs of the following functions.(a)f(x) = 3−2x(−1< x <4)(b)f(x) =−12x2−2x−5(x≥ −5)You can use the graph of a function to visualise its domain on thehorizontal axis. The domain consists of all the possible input numbers ofthe function, that is, all points on the horizontal axis that lie directlybelow or above a point on the graph, as illustrated in Figure 16.xyFigure 16The domain of a function marked on the horizontal axisActivity 13Identifying the domains of functionsWrite down the domains of the functions whose graphs are shown below,using interval notation. All the endpoints of the intervals involved areintegers, and in part (b) the graph continues indefinitely to the left andright.(a)xy1234−11(b)xy−5−4−3−2−11212As you’ve seen, a function has exactly one output number for every inputnumber. So if you draw the vertical line through any number in thedomain of a function on the horizontal axis, then it will cross the graph of221

Unit 3Functionsthe functionexactly once, as illustrated in Figure 17(a). If you can draw avertical line that crosses a curve more than once, then the curve isn’t thegraph of a function. For example, the curve in Figure 17(b) isn’t the graphof a function.xy(a)xy(b)Figure 17(a) The graph of a function (b) a curve that isn’t the graph ofa functionActivity 14Identifying graphs of functionsWhich of the following diagrams are the graphs of functions?(a)xy(b)xy(c)xy(d)xy(e)xy(f)xy(g)xy(h)xy(i)xy(j)xy(k)xy(l)xy222

1Functions and their graphsIncreasing and decreasing functionsFigure 18 shows the graph of a function with domain [−1,9]. Asxincreases, the graph first slopes up, then slopes down, then slopes upagain. It changes from sloping up to sloping down whenx= 2, and itchanges from sloping down to sloping up again whenx= 6.xy−3−2−1123456789101112−2−1123456Figure 18The graph of a functionTo express these facts about the functionf, we say thatfisincreasing onthe interval[−1,2],decreasing on the interval[2,6], and increasing againon the interval [6,9]. Here are the formal definitions of these terms. Thedefinitions are illustrated in Figure 19.Functions increasing or decreasing on an intervalA functionfisincreasing on the intervalIif for all valuesx1andx2inIsuch thatx1< x2,f(x1)< f(x2).A functionfisdecreasing on the intervalIif for all valuesx1andx2inIsuch thatx1< x2,f(x1)> f(x2).(The intervalImust be part of the domain off.)223

Unit 3Functionsx1x2f(x1)f(x2)xyy=f(x)I(a)x1x2f(x1)f(x2)xyy=f(x)I(b)Figure 19(a) A function increasing on an intervalI(b) a functiondecreasing on an intervalIFor example, the functionf(x) =x2, whose graph is shown in Figure 20, isxy−3−2−11231234Figure 20The graph of thefunctionf(x) =x2decreasing on the interval (−∞,0] and increasing on the interval [0,∞).

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Complex number, John Venn