StewartCalcET8-01-01.ppt

Copyright Cengage Learning.All rights reserved.1Functions and ModelsCopyright Cengage Learning.All rights reserved.1.1Four Ways to Represent a Function3Four Ways to Represent a FunctionFunctions arise whenever one quantity depends onanother.Consider the following four situations.A.The area A of a circle depends on the radius r of the circle.The rule that connects r and A is given by the equation A=r 2.With each positive number r there is associated one value of A,and we say that A is a function of r.4Four Ways to Represent a FunctionB.The human population of the world P depends on the time t.The table gives estimates of the world population P(t)at time t,for certain years.For instance,P(1950)2,560,000,000 But for each value of the time tthere is a corresponding value of P,and we say that P is a function of t.5Four Ways to Represent a FunctionC.The cost C of mailing an envelope depends on its weight w.Although there is no simple formula that connects w and C,the post office has a rule for determining C when w is known.D.The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t.6Four Ways to Represent a Function Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994.For a given value of t,the graph provides a corresponding value of a.Vertical ground acceleration during the Northridge earthquakeFigure 17Four Ways to Represent a FunctionWe usually consider functions for which the sets D and E are sets of real numbers.The set D is called the domain of the function.The number f(x)is the value of f at x and is read“f of x.”The range of f is the set of all possible values of f(x)as x varies throughout the domain.A symbol that represents an arbitrary number in the domain of a function f is called an independent variable.8Four Ways to Represent a FunctionA symbol that represents a number in the range of f is called a dependent variable.In Example A,for instance,r is the independent variable and A is the dependent variable.Its helpful to think of a function as a machine(see Figure 2).Machine diagram for a function fFigure 29Four Ways to Represent a FunctionIf x is in the domain of the function f,then when x enters the machine,its accepted as an input and the machine produces an output f(x)according to the rule of the function.Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs.10Four Ways to Represent a FunctionAnother way to picture a function is by an arrow diagram as in Figure 3.Each arrow connects an element of D to an element of E.The arrow indicates that f(x)is associated with x,f(a)is associated with a,and so on.Arrow diagram for fFigure 311Four Ways to Represent a FunctionThe most common method for visualizing a function is its graph.If f is a function with domain D,then its graph is the set of ordered pairs (x,f(x)|x DIn other words,the graph of f consists of all points(x,y)in the coordinate plane such that y=f(x)and x is in the domain of f.The graph of a function f gives us a useful picture of the behavior or“life history”of a function.12Four Ways to Represent a FunctionSince the y-coordinate of any point(x,y)on the graph is y=f(x),we can read the value of f(x)from the graph as being the height of the graph above the point x(see Figure 4).Figure 413Four Ways to Represent a FunctionThe graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5.Figure 514Example 1The graph of a function f is shown in Figure 6.(a)Find the values of f(1)and f(5).(b)What are the domain and range of f?Figure 615Example 1 Solution (a)We see from Figure 6 that the point(1,3)lies on the graph of f,so the value of f at 1 is f(1)=3.(In other words,the point on the graph that lies above x=1 is 3 units above the x-axis.)When x=5,the graph lies about 0.7 unit below the x-axis,so we estimate that f(5)0.7.(b)We see that f(x)is defined when 0 x 7,so the domain of f is the closed interval 0,7.Notice that f takes on all values from 2 to 4,so the range of f is y|2 y 4=2,416Representations of Functions17Representations of FunctionsThere are four possible ways to represent a function:verbally (by a description in words)numerically (by a table of values)visually (by a graph)algebraically (by an explicit formula)18Example 4When you turn on a hot-water faucet,the temperature T of the water depends on how long the water has been running.Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.Solution:The initial temperature of the running water is close to room temperature because the water has been sitting in the pipes.When the water from the hot-water tank starts flowing from the faucet,T increases quickly.In the next phase,T is constant at the temperature of the heated water in the tank.19Example 4 Solution When the tank is drained,T decreases to the temperature of the water supply.This enables us to make the rough sketch of T as a function of t in Figure 11.Figure 11contd20Representations of FunctionsThe graph of a function is a curve in the xy-plane.But the question arises:Which curves in the xy-plane are graphs of functions?This is answered by the following test.21Representations of FunctionsThe reason for the truth of the Vertical Line Test can be seen in Figure 13.Figure 13(b)This curve doesnt represent a function.(a)This curve represents a function.22Representations of FunctionsIf each vertical line x=a intersects a curve only once,at (a,b),then exactly one functional value is defined byf(a)=b.But if a line x=a intersects the curve twice,at(a,b)and(a,c),then the curve cant represent a function because a function cant assign two different values to a.23Representations of FunctionsFor example,the parabola x=y2 2 shown in Figure 14(a)is not the graph of a function of x because,as you can see,there are vertical lines that intersect the parabola twice.The parabola,however,does contain the graphs of two functions of x.x=y2 2Figure 14(a)24Representations of FunctionsNotice that the equation x=y2 2 implies y2=x+2,so .Thus the upper and lower halves of the parabola are the graphs of the functions andSee Figures 14(b)and(c).Figure 14(c)Figure 14(b)25Representations of FunctionsWe observe that if we reverse the roles of x and y,then the equation x=h(y)=y2 2 does define x as a function of y(with y as the independent variable and x as the dependent variable)and the parabola now appears as the graph of the function h.26Piecewise Defined Functions27Example 7A function f is defined by1 x if x 1x2 if x 1Evaluate f(2),f(1),and f(0)and sketch the graph.Solution:Remember that a function is a rule.For this particular function the rule is the following:First look at the value of the input x.If it happens thatx 1,then the value of f(x)is 1 x.f(x)=28Example 7 Solution On the other hand,if x 1,then the value of f(x)is x2.Since 2 1,we have f(2)=1 (2)=3.Since 1 1,we have f(1)=1 (1)=2.Since 0 1,we have f(0)=02=0.How do we draw the graph of f?We observe that if x 1,then f(x)=1 x,so the part of the graph of f that lies to the left of the vertical line x=1 must coincide with the line y=1 x,which has slope 1 and y-intercept 1.contd29Example 7 Solution If x 1,then f(x)=x2,so the part of the graph of f that lies to the right of the line x=1 must coincide with the graph of y=x2,which is a parabola.This enables us to sketch the graph in Figure 15.The solid dot indicates that the point(1,2)is included on the graph;the open dot indicates that the point(1,1)is excluded from the graph.Figure 15contd30Piecewise Defined FunctionsThe next example of a piecewise defined function is the absolute value function.Recall that the absolute value of a number a,denoted by|a|,is the distance from a to 0 on the real number line.Distances are always positive or 0,so we have|a|0 for every number aFor example,|3|=3|3|=3|0|=0|1|=1|3|=331Piecewise Defined FunctionsIn general,we have(Remember that if a is negative,then a is positive.)32Example 8 Sketch the graph of the absolute value function f(x)=|x|.Solution:From the preceding discussion we know that x if x 0|x|=x if x 033Example 8 SolutionUsing the same method as in Example 7,we see that the graph of f coincides with the line y=x to the right of the y-axis and coincides with the line y=x to the left of the y-axis(see Figure 16).Figure 16contd34Example 10In Example C at the beginning of this section we considered the cost C(w)of mailing a large envelope with weight w.In effect,this is a piecewise defined function because,from the table shown in the right side,we have 0.98 if 0 w 1 1.19 if 1 w 2C(w)=1.40 if 2 w 3 1.61 if 3 w 435Example 10The graph is shown in Figure 18.You can see why functions similar to this one are called step functionsthey jump from one value to the next.Figure 18contd36Symmetry37SymmetryIf a function f satisfies f(x)=f(x)for every number x in its domain,then f is called an even function.For instance,the function f(x)=x2 is even because f(x)=(x)2=x2=f(x)The geometric significance of aneven function is that its graph is symmetric with respect to the y-axis(see Figure 19).Figure 19An even function38SymmetryThis means that if we have plotted the graph of f for x 0,we obtain the entire graph simply by reflecting this portion about the y-axis.If f satisfies f(x)=f(x)for every number x in its domain,then f is called an odd function.For example,the functionf(x)=x3 is odd because f(x)=(x)3=x3=f(x)39SymmetryThe graph of an odd function is symmetric about the origin(see Figure 20).If we already have the graph of f for x 0,we can obtain the entire graph by rotating this portion through 180 about the origin.Figure 20An odd function40Example 11Determine whether each of the following functions is even,odd,or neither even nor odd.(a)f(x)=x5+x (b)g(x)=1 x4 (c)h(x)=2x x2 Solution:(a)f(x)=(x)5+(x)=(1)5x5+(x)=x5 x=(x5+x)=f(x)Therefore f is an odd function.41Example 11 Solution (b)g(x)=1 (x)4=1 x4=g(x)So g is even.(c)h(x)=2(x)(x)2=2x x2 Since h(x)h(x)and h(x)h(x),we conclude that h is neither even nor odd.contd42SymmetryThe graphs of the functions in Example 11 are shown in Figure 21.Notice that the graph of h is symmetric neither about the y-axis nor about the origin.Figure 21(b)(c)(a)43Increasing and Decreasing Functions44Increasing and Decreasing FunctionsThe graph shown in Figure 22 rises from A to B,falls from B to C,and rises again from C to D.The function f is said to be increasing on the interval a,b,decreasing on b,c,and increasing again on c,d.Figure 2245Increasing and Decreasing FunctionsNotice that if x1 and x2 are any two numbers betweena and b with x1 x2,then f(x1)f(x2).We use this as the defining property of an increasing function.46Increasing and Decreasing FunctionsIn the definition of an increasing function it is important to realize that the inequality f(x1)f(x2)must be satisfied for every pair of numbers x1 and x2 in I with x1 x2.You can see from Figure 23that the function f(x)=x2 is decreasing on the interval(,0and increasing on the interval0,).Figure 23