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4 Absolute Value FunctionsIn this chapter we will introduce the absolute value function, one of the more usefulfunctions that we will study in this course. Because it is closely related to the conceptof distance, it is a favorite among statisticians, mathematicians, and other practitionersof science.Most readers probably already have an intuitive understanding of absolute value.You’ve probably seen that the absolute value of seven is seven, i.e., 7 7, and theabsolute value of negative seven is also seven, i.e., 7 7. That is, the absolutevalue function takes a number as input, and then makes that number positive (if itisn’t already). Technically, because 0 0, which is not a positive number, we areforced to say that the absolute value function takes a number as input, and then makesit nonnegative (positive or zero).However, as you advance in your coursework, you will quickly discover that thisintuitive notion of absolute value is insufficient when tackling more sophisticated problems. In this chapter, as we try to raise our understanding of absolute value to a higherplane, we will encounter piecewise-defined functions and use them to create piecewisedefinitions for absolute value functions. These piecewise definitions will help us drawthe graphs of a variety of absolute value functions.Finally we’ll conclude our work in this chapter by developing techniques for solvingequations and inequalities containing expressions that implement the absolute valuefunction.Table of Contents4.14.24.34.4Piecewise-Defined Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Piecewise Constant FunctionsPiecewise-Defined FunctionsExercisesAnswersAbsolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Piecewise Definition of Absolute ValueConstructing Piecewise DefinitionsDrawing the Graph of an Absolute Value FunctionUsing TransformationsExercisesAnswersAbsolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Solving x aProperties of Absolute ValueDistance RevisitedExercisesAnswersAbsolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Solving x 82385388391391
3344.5Chapter 4Absolute Value FunctionsSolving x aRevisiting DistanceExercisesAnswersIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397403405408413CopyrightAll parts of this intermediate algebra textbook are copyrighted in the name ofDepartment of Mathematics, College of the Redwoods. They are not in the publicdomain. However, they are being made available free for use in educational institutions. This offer does not extend to any application that is made for profit.Users who have such applications in mind should contact David Arnold at davidarnold@redwoods.edu or Bruce Wagner at bruce-wagner@redwoods.edu.This work (including all text, Portable Document Format files, and any other original works), except where otherwise noted, is licensed under a Creative CommonsAttribution-NonCommercial-ShareAlike 2.5 License, and is copyrighted C 2006,Department of Mathematics, College of the Redwoods. To view a copy of thislicense, visit http://creativecommons.org/licenses/by-nc-sa/2.5/ or send a letterto Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California,94105, USA.Version: Fall 2007
Section 4.1Piecewise-Defined Functions3354.1 Piecewise-Defined FunctionsIn preparation for the definition of the absolute value function, it is extremely importantto have a good grasp of the concept of a piecewise-defined function. However, beforewe jump into the fray, let’s take a look at a special type of function called a constantfunction.One way of understanding a constant function is to have a look at its graph.I Example 1.Sketch the graph of the constant function f (x) 3.Because the notation f (x) 3 is equivalent to the notation y 3, we can sketch agraph of f by drawing the graph of the horizontal line having equation y 3, as shownin Figure 1.y10f (x) 310xFigure 1. The graph of a constant function is a horizontal line.When you look at the graph in Figure 1, note that every point on the horizontal linehaving equation f (x) 3 has a y-value equal to 3. We say that the y-values on thishorizontal line are constant, for the simple reason that they are constantly equal to 3.The function form works in precisely the same manner. Look again at the notationf (x) 3.Note that no matter what number you substitute for x in the left-hand side of f (x) 3,the right-hand side is constantly equal to 3. Thus, f ( 5) 3, f ( 1/2) 3, f ( 2) 3, or f (π) 3.The above discussion leads to the following definition.1Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/Version: Fall 2007
336Chapter 4Absolute Value FunctionsDefinition 2. The function defined by f (x) c, where c is a constant (fixedreal number), is called a constant function.Two comments are in order:1. f (x) c for all real numbers x.2. The graph of f (x) c is a horizontal line. It consists of all the points (x, y) havingy-value equal to c.Piecewise Constant FunctionsPiecewise functions are a favorite of engineers. Let’s look at an example.I Example 3. Suppose that a battery provides no voltage to a circuit when a switchis open. Then, starting at time t 0, the switch is closed and the battery providesa constant 5 volts from that time forward. Create a piecewise function modeling theproblem constraints and sketch its graph.This is a fairly simple exercise, but we will have to introduce some new notation.First of all, if the time t is less than zero (t 0), then the voltage is 0 volts. If thetime t is greater than or equal to zero (t 0), then the voltage is a constant 5 volts.Here is the notation we will use to summarize this description of the voltage. 0, if t 0,V (t) (4)5, if t 0Some comments are in order: The voltage difference provide by the battery in the circuit is a function of time.Thus, V (t) represents the voltage at time t.The notation used in (4) is universally adopted by mathematicians in situationswhere the function changes definition depending on the value of the independentvariable. This definition of the function V is called a “piecewise definition.” Becauseeach of the pieces in this definition is constant, the function V is called a piecewiseconstant function.This particular function has two pieces. The function is the constant functionV (t) 0, when t 0, but a different constant function, V (t) 5, when t 0.If t 0, V (t) 0. For example, for t 1, t 10, and t 100,V ( 1) 0,V ( 10) 0,andV ( 100) 0.On the other hand, if t 0, then V (t) 5. For example, for t 0, t 10, and t 100,V (0) 5,V (10) 5,andV (100) 5.Before we present the graph of the piecewise constant function V , let’s pause for amoment to make sure we understand some standard geometrical terms.Version: Fall 2007
Section 4.1Piecewise-Defined Functions337Geometrical Terms. A line stretches indefinitely in two directions, as shown in Figure 2(a).If a line has a fixed endpoint and stretches indefinitely in only one direction,as shown in Figure 2(b), then it is called a ray. If a portion of the line is fixed at each end, as shown in Figure 2(c), then itis called a line segment.BABBAA(a)(b)Figure 2.(c)Lines, rays, and segments.With these terms in hand, let’s turn our attention to the graph of the voltage definedby equation (4). When t 0, then V (t) 0. Normally, the graph of V (t) 0 wouldbe a horizontal line where each point on the line has V -value equal to zero. However,V (t) 0 only if t 0, so the graph is the horizontal ray that starts at the origin, thenmoves indefinitely to the left, as shown in Figure 3. That is, the horizontal line V 0has been restricted to the domain {t : t 0} and exists only to the left of the origin.Similarly, when t 0, then V (t) 5 is the horizontal ray shown in Figure 3. Eachpoint on the ray has a V -value equal to 5.V1050Figure 3.of time t.10tThe voltage as a functionVersion: Fall 2007
338Chapter 4Absolute Value FunctionsTwo comments are in order: Because V (t) 0 only when t 0, the point (0, 0) is unfilled (it is an open circle).The open circle at (0, 0) is a mathematician’s way of saying that this particularpoint is not plotted or shaded.Because V (t) 5 when t 0, the point (0, 5) is filled (it is a filled circle). Thefilled circle at (0, 5) is a mathematician’s way of saying that this particular point isplotted or shaded.Let’s look at another example.I Example 5.Consider the piecewise-defined function( 0, if x 0,f (x) 1, if 0 x 2,2, if x 2.(6)Evaluate f (x) at x 1, 0, 1, 2, and 3. Sketch the graph of the piecewise function f .Because each piece of the function in (6) is constant, evaluation of the function ispretty easy. You just have to select the correct piece. Note that x 1 is less than 0, so we use the first piece and write f ( 1) 0.Note that x 0 satisfies 0 x 2, so we use the second piece and write f (0) 1.Note that x 1 satisfies 0 x 2, so we use the second piece and write f (1) 1.Note that x 2 satisfies x 2, so we use the third piece and write f (2) 2.Finally, note that x 3 satisfies x 2, so we use the third piece and write f (3) 2.The graph is just as simple to sketch. Because f (x) 0 for x 0, the graph of this piece is a horizontal ray with endpointat x 0. Each point on this ray will have a y-value equal to zero and the ray willlie entirely to the left of x 0, as shown in Figure 4. Because f (x) 1 for 0 x 2, the graph of this piece is a horizontal segmentwith one endpoint at x 0 and the other at x 2. Each point on this segment willhave a y-value equal to 1, as shown in Figure 4. Because f (x) 2 for x 2, the graph of this piece is a horizontal ray with endpointat x 2. Each point on this ray has a y-value equal to 2 and the ray lies entirelyto the right of x 2, as shown in Figure 4.Several remarks are in order: The function is zero to the left of the origin (for x 0), but not at the origin. Thisis indicated by an empty circle at the origin, an indication that we are not plottingthat particular point.For 0 x 2, the function equals 1. That is, the function is constantly equal to 1for all values of x between 0 and 2, including zero but not including 2. This is whyyou see a filled circle at (0, 1) and an empty circle at (2, 1).Finally, for x 2, the function equals 2. That is, the function is constantly equalto 2 whenever x is greater than or equal to 2. That is why you see a filled circle at(2, 2).Version: Fall 2007
Section 4.1Piecewise-Defined Functions339y1010xFigure 4. Sketching the graph of thepiecewise function (6).Piecewise-Defined FunctionsNow, let’s look at a more generic situation involving piecewise-defined functions—onewhere the pieces are not necessarily constant. The best way to learn is by doing, solet’s start with an example.I Example 7.Consider the piecewise-defined function x 2, if x 2,f (x) x 2,if x 2.(8)Evaluate f (x) for x 0, 1, 2, 3 and 4, then sketch the graph of the piecewise-definedfunction.The function changes definition at x 2. If x 2, then f (x) x 2. Becauseboth 0 and 1 are strictly less than 2, we evaluate the function with this first piece ofthe definition.f (x) x 2andf (x) x 2f (0) 0 2f (1) 1 2f (0) 2f (1) 1.On the other hand, if x 2, then f (x) x 2. Because 2, 3, and 4 are all greaterthan or equal to 2, we evaluate the function with this second piece of the definition.f (x) x 2andf (x) x 2andf (x) x 2f (2) 2 2f (3) 3 2f (4) 4 2f (2) 0f (3) 1f (4) 2.Version: Fall 2007
340Chapter 4Absolute Value FunctionsOne possible approach to the graph of f is to place the points we’ve already calculated, plus a couple extra, in a table (see Figure 5(a)), plot them (see Figure 5(b)),then intuit the shape of the graph from the evidence provided by the plotted points.This is done in Figure 5(c).x 1012345f (x)3210123(a)Figure 5.y10y10f10(b)x10(c)Plotting the graph of the piecewise function defined in (8).However pragmatic, this point-plotting approach is a bit tedious; but, more importantly, it does not provide the background necessary for the discussion of the absolutevalue function in the next section. We need to stretch our understanding to a higherlevel. Fortunately, all the groundwork is in place. We need only apply what we alreadyknow about the equations of lines to fit this piecewise situation.Alternative approach. Let’s use our knowledge of the equation of a line (i.e.y mx b) to help sketch the graph of the piecewise function defined in (8).Let’s sketch the first piece of the function f defined in (8). We have f (x) x 2,provided x 2. Normally, this would be a line (with slope 1 and intercept 2), but weare to sketch only a part of that line, the part where x 2 (x is to the left of 2). Thus,this piece of the graph will be a ray, starting at the point where x 2, then movingindefinitely to the left.The easiest way to sketch a ray is to first calculate and plot its fixed endpoint (inthis case at x 2), then plot a second point on the ray having x-value less than 2, thenuse a ruler to draw the ray.With this thought in mind, to find the coordinates of the endpoint of the ray,substitute x 2 in f (x) x 2 to get f (2) 0. Now, technically, we’re notsupposed to use this piece of the function unless x is strictly less than 2, but we coulduse it with x 1.9, or x 1.99, or x 1.999, etc. So let’s go ahead and use x 2in this piece of the function, but indicate that we’re not actually supposed to use thispoint by drawing an “empty circle” at (2, 0), as shown in Figure 6(a).To complete the plot of the ray, we need a second point that lies to the left of itsendpoint at (2, 0). Note that x 0 is to the left of x 2. Evaluate f (x) x 2at x 0 to obtain f (0) 0 2 2. This gives us the second point (0, 2), which weplot as shown in Figure 6(a). Finally, draw the ray with endpoint at (2, 0) and secondpoint at (0, 2), as shown in Figure 6(a).Version: Fall 2007x
Section 4.1Piecewise-Defined Functionsy10341y10(0, 2)(4, 2)10(2, 0)x(2, 0)(a)10x(b)Figure 6.Sketch each piece separately.We now repeat this process for the second piece of the function defined in (8). Theequation of the second piece is f (x) x 2, provided x 2. Normally, f (x) x 2would be a line (with slope 1 and intercept 2), but we’re only supposed to sketch thatpart of the line that lies to the right of or at x 2. Thus, the graph of this second pieceis a ray, starting at the point with x 2 and continuing to the right. If we evaluatef (x) x 2 at x 2, then f (2) 2 2 0. Thus, the fixed endpoint of the ray is atthe point (2, 0). Since we’re actually supposed to use this piece with x 2, we indicatethis fact with a filled circle at (2, 0), as shown in Figure 6(b).We need a second point to the right of this fixed endpoint, so we evaluate f (x) x 2at x 4 to get f (4) 4 2 2. Thus, a second point on the ray is the point (4, 2).Finally, we simply draw the ray, starting at the endpoint (2, 0) and passing throughthe second point at (4, 2), as shown in Figure 6(b).To complete the graph of the piecewise function f defined in equation (8), simplycombine the two pieces in Figure 6(a) and Figure 6(b) to get the finished graphin Figure 7. Note that the graph in Figure 7 is identical to the earlier result inFigure 5(c).Let’s try this alternative procedure in another example.I Example 9.definitionA source provides voltage to a circuit according to the piecewise V (t) 0, if t 0,t, if t 0.(10)Sketch the graph of the voltage V versus time t.For all time t that is less than zero, the voltage V is zero. The graph of V (t) 0 is aconstant function, so its graph is normally a horizontal line. However, we must restrictVersion: Fall 2007
342Chapter 4Absolute Value Functionsy10f10xFigure 7. Combiningboth pieces.the graph to the domain ( , 0), so this piece of equation (10) will be a horizontalray, starting at the origin and moving indefinitely to the left, as shown in Figure 8(a).On the other hand, V (t) t for all values of t that are greater than or equal tozero. Normally, this would be a line with slope 1 and intercept zero. However, we mustrestrict the domain to [0, ), so this piece of equation (10) will be a ray, starting atthe origin and moving indefinitely to the right. The endpoint of this ray starts at t 0. Because V (t) t, V (0) 0. Hence, theendpoint of this ray is at the point (0, 0).Choose any value of t that is greater than zero. We’ll choose t 5. BecauseV (t) t, V (5) 5. This gives us a second point on the ray at (5, 5), as shown inFigure 8(b).V10V10(5, 5)(0, 0)10t(0, 0)10t(b) V (t) t for t 0.(a) V (t) 0 for t 0.Figure 8.Finally, to provide a complete graph of the voltage function defined by equation (10),we combine the graphs of each piece of the definition shown in Figures 8(a) and (b).Version: Fall 2007
Section 4.1Piecewise-Defined Functions343The result is shown in Figure 9. Engineers refer to this type of input function as a“ramp function.”V10V10tFigure 9. The graph of the rampfunction defined by equation (10).Let’s look at a very practical application of piecewise functions.I Example 11. The federal income tax rates for a single filer in the year 2005 aregiven in Table 1.IncomeTax RateUp to 7,15010% 7,151- 29,05015% 29,051- 70,35025% 70,351- 146,75028% 146,751- 319,100 33% 319,101 or more35%Table 1. 2005 Federal Income Taxrates for single filer.Create a piecewise definition that provides the tax rate as a function of personal income.In reporting taxable income, amounts are rounded to the nearest dollar on thefederal income tax form. Technically, the domain is discrete. You can report a taxableincome of 35,000 or 35,001, but numbers between these two incomes are not usedon the federal income tax form. However, we will think of the income as a continuum,allowing the income to be any real number greater than or equal to zero. If we didnot do this, then our graph would be a series of dots–one for each dollar amount. Wewould have to plot lots of dots!Version: Fall 2007
344Chapter 4Absolute Value FunctionsWe will let R represent the tax rate and I represent the income. The goal is todefine R as a function of I. If income I is any amount greater than or equal to zero, and less than or equalto 7,150, the tax rate R is 10% (i.e., R 0.10). Thus, if 0 I 7, 150,R(I) 0.10. If income I is any amount that is strictly greater than 7,150 but less than or equal to 29,050, then the tax rate R is 15% (i.e., R 0.15). Thus, if 7, 150 I 29, 050,then R(I) 0.15.Continuing in this manner, we can construct a piecewise definition of rate R as afunction of taxable income I. 0.10, if 0 I 7, 150, 0.15, if 7, 150 I 29, 050, 0.25, if 29, 050 I 70, 350,(12)R(I) 0.28, if 70, 350 I 146, 750, 0.33, if 146, 750 I 319, 100,0.35, if I 319, 100.Let’s turn our attention to the graph of this piecewise-defined function. All ofthe pieces are constant functions, so each piece will be a horizontal line at a levelindicating the tax rate. However, each of the first five pieces of the function defined inequation (12) are segments, because the rate is defined on an interval with a startingand ending income. The sixth and last piece is a ray, as it has a starting endpoint, butthe rate remains constant for all incomes above 319,100. We use this knowledge toconstruct the graph shown in Figure 10.The first rate is 10% and this is assigned to taxable income starting at 0 and endingat 7,150, inclusive. Thus, note the first horizontal line segment in Figure 10 thatruns from 0 to 7,150 at a height of R 0.10. Note that each of the endpoints arefilled circles.The second rate is 15% and this is assigned to taxable incomes greater than 7,150,but less than or equal to 29,050. The second horizontal line segment in Figure 10runs from 7,150 to 29,050 at a height of R 0.15. Note that the endpoint at the leftend of this horizontal segment is an open circle while the endpoint on the right end isa filled circle because the taxable incomes range on 7, 150 I 29, 050. Thus, weexclude the left endpoint and include the right endpoint.The remaining segments are drawn in a similar manner.The last piece assigns a rate of R 0.35 to all taxable incomes strictly above 319,100. Hence, the last piece is a horizontal ray, starting at ( 319 100, 0.35) andextending indefinitely to the right. Note that the left endpoint of this ray is an opencircle because the rate R 0.35 applies to taxable incomes I 319, 100.Let’s talk a moment about the domain and range of the function R defined byequation (12). The graph of R is depicted in Figure 10. If we project all points onthe graph onto the horizontal axis, the entire axis will “lie in shadow.” Thus, at firstVersion: Fall 2007
Section 4.1Piecewise-Defined Functions345R0.400.300.200.10100, 000Figure 10.200, 000300, 000I400, 000The graph of the tax rate R versus taxable income I.glance, one would state that the domain of R is the set of all real numbers that aregreater than or equal to zero.However, remember that we chose to model a discrete situation with a continuum.Taxable income is always rounded to the nearest dollar on federal income tax forms.Therefore, the domain is actually all whole numbers greater than or equal to zero. Insymbols,Domain {I W : I 0}.To find the range of R, we would project all points on the graph of R in Figure 10onto the vertical axis. The result would be that six points would be shaded on thevertical axis, one each at 0.10, 0.15, 0.25, 0.28, 0.33, and 0.35. Thus, the range is afinite discrete set, so it’s best described by simply listing its members.Range {0.10, 0.15, 0.25, 0.28, 0.33, 0.35}Version: Fall 2007
346Chapter 4Version: Fall 2007Absolute Value Functions
Section 4.1Piecewise-Defined Functions3474.1 Exercises1. Given the function defined by therule f (x) 3, evaluate f ( 3), f (0) andf (4), then sketch the graph of f .2. Given the function defined by therule g(x) 2, evaluate g( 2), g(0) andg(4), then draw the draw the graph of g.3. Given the function defined by therule h(x) 4, evaluate h( 2), h(a),and h(2x 3), then draw the graph of h.4. Given the function defined by therule f (x) 2, evaluate f (0), f (b), andf (5 4x), then draw the graph of f .5. The speed of an automobile traveling on the highway is a function of timeand is described by the constant function v(t) 30, where t is measured inhours and v is measured in miles perhour. Draw the graph of v versus t. Besure to label each axis with the appropriate units. Shade the area under thegraph of v over the time interval [0, 5]hours. What is the area under the graphof v over this time interval and what doesit represent?6. The speed of a skateboarder as shetravels down a slope is a function of timeand is described by the constant functionv(t) 8, where t is measured in secondsand v is measured in feet per second.Draw the graph of v versus t. Be sureto label each axis with the appropriateunits. Shade the area under the graph ofv over the time interval [0, 60] seconds.What is the area under the graph of vover this time interval and what does itrepresent?27. An unlicensed plumber charges 15dollars for each hour of labor. Let’s define this rate as a function of time byr(t) 15, where t is measured in hoursand r is measured in dollars per hour.Draw the graph of r versus t. Be sure tolabel each axis with appropriate units.Shade the area under the graph of r overthe time interval [0, 4] hours. What isarea under the graph of r over this timeinterval and what does it represent?8. A carpenter charges a fixed rate foreach hour of labor. Let’s describe thisrate as a function of time by r(t) 25,where t is measured in hours and r ismeasured in dollars per hour. Draw thegraph of r versus t. Be sure to label eachaxis with appropriate units. Shade thearea under the graph of r over the timeinterval [0, 5] hours. What is the area under the graph of r over this time intervaland what does it represent?9. Given the function defined by therule 0, if x 0f (x) 2, if x 0,evaluate f ( 2), f (0), and f (3), then drawthe graph of f on a sheet of graph paper.State the domain and range of f .10. Given the function defined by therule 2, if x 0f (x) 0, if x 0,evaluate f ( 2), f (0), and f (3), then drawthe graph of f on sheet of graph paper.State the domain and range of f .Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/Version: Fall 2007
348Chapter 4Absolute Value Functions11. Given the function defined by therule( 3, if x 2,g(x) 14.5if 2 x 2,if x 2,1,3,yfevaluate g( 3), g( 2), and g(5), thendraw the graph of g on a sheet of graphpaper. State the domain and range of g.5x12. Given the function defined by therule( 4,if x 1,g(x) 2,if 1 x 2, 3, if x 2,15.5evaluate g( 1), g(2), and g(3), then drawthe graph of g on a sheet of graph paper.State the domain and range of g.ygIn Exercises 13-16, determine a piecewise definition of the function describedby the graphs, then state the domain andrange of the function.5x13.5y16.f55yx5gVersion: Fall 2007x
Section 4.117.Given the piecewise definition x 3, if x 3,f (x) x 3,if x 3,evaluate f ( 4) and f (0), then draw thegraph of f on a sheet of graph paper.State the domain and range of the function.18.Given the piecewise definition x 1, if x 1,f (x) x 1,if x 1,evaluate f ( 2) and f (3), then draw thegraph of f on a sheet of graph paper.State the domain and range of the function.19.Piecewise-Defined Functions349by the battery drops immediately to zerovolts. Sketch a graph of the voltage vversus time t, label each axis with theappropriate units, then provide a piecewise definition of the voltage v suppliedby the battery as a function of time t.22. Prior to time t 0 minutes, a drumis empty. At time t 0 minutes a hoseis turned on and the water level in thedrum begins to rise at a constant rateof 2 inches every minute. Let h represent water level (in inches) at time t (inminutes). Sketch the graph of h versust, label the axes with appropriate units,then provide a piecewise definition of has a function of t.Given the piecewise definition 2x 3, if x 3/2,g(x) 2x 3,if x 3/2,evaluate g(0) and g(3), then draw thegraph of g on a sheet of graph paper.State the domain and range of the function.20.Given the piecewise definition 3x 4, if x 4/3,g(x) 3x 4,if x 4/3,evaluate g( 2) and g(3), then draw thegraph of g on a sheet of graph paper.State the domain and range of the function.21. A battery supplies voltage to anelectric circuit in the following manner.Before time t 0 seconds, a switch isopen, so the voltage supplied by the battery is zero volts. At time t 0 seconds,the switch is closed and the battery begins to supply a constant 3 volts to thecircuit. At time t 2 seconds, the switchis opened again, and the voltage suppliedVersion: Fall 2007
350Chapter 4Absolute Value Functions4.1 Answers1.f ( 3) 3, f (0) 3, and f (4) 3.5y7. The area under the curve is 150 miles.This is the distance traveled by the car.r (dollars/h)f (x) 3r(t) 15155x03. h( 2) 4, h(a) 4, and h(2x 3) 4.59.0t (h)4f ( 2) 0, f (0) 2, and f (3) 2.5yyf55xxh(x) 4The domain of f is the set of all realnumbers. The range of f is {0, 2}.5. The area under the curve is 150 miles.This is the distance traveled by the car.11.3.g( 3) 3, g( 2) 1, and g(5) v (mi/h)5v(t) 3030yg500Version: Fall 20075t (h)x
Section 4.1The domain of g is all real numbers. Therange of g is { 3, 1, 3}.Piecewise-Defined Functions21.The graph follows.513. f (x) 351V (volts)3,if x 0, 2, if x 0.Domain of f is the set of all real numbers.The range of f is { 2, 3}.5t (s)15.if x 0, 2, if 0 x 2,2,if x 2.( 2,g(x) The domain of f is the set of all realnumbers. The range of f is { 2, 2}.The piecewise definition is( 0, if t 0,v(t) 17.f ( 4) 1 and f (0) 3.5y3, if 0 t 2,0, if t 2.f(0,3)( 4,1)5( 3,0)xThe domain of f is the set of all realnumbers. The range of f is {y : y 0}.19.g( 2) 7 and g(2) 1.5yg(3,3)(0,3)(3/2,0)5xThe domain of g is the set of all real numbers. The range of g is {y : y 0}.Version: Fall 2007
352Chapter 4Version: Fall 2007Absolute Value Functions
Section 4.2Absolute Value 3534.2 Absolute ValueNow that we have the fundamentals of piecewise-defined functions in place, we areready to introduce the absolute value function. First, let’s state a trivial reminder ofwhat it means to take the absolute value of a real number.In a sense, the absolute value of a number is a measure of its magnitude, sans(without) its sign. Thus, 7 7 7] 7.and(1)Here is the formal definition of the absolute value of a real number.Definition 2. To find the absolute value of any real number, first locate thenumber on the real line. x x0The absolute value of the number is defined as its distance from the origin.For example, to find the absolute value of 7, locate 7 on the real line and then findits distance from the origin. 7 707To find the absolute value of 7, locate 7 on the real line and then find its distancefrom the origin. 7 7 70Some like to say that taking the absolute value “produces a number that is alwayspositive.” However, this ignores an important exception, that is, 0 0.3(3)Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/Version: Fall 2007
354Chapter 4Absolute Value FunctionsThus, the correct statement is “the absolute
336 Chapter 4 Absolute Value Functions Version: Fall2007 Denition 2. The function dened by f(x) c, where cis a constant (xed realnumber),iscalleda constantfunction. Twocommentsareinorder: 1. f(x) cforallrealnumbersx. 2. Thegraphoff(x) cisahorizontalline.Itconsistsofallthepoints(x,y) havingy-valueequaltoc. PiecewiseConstantFunctions
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ANSI A300 Purpose: To provide performance standards for developing written specifications for tree management. Currently nine individual parts . ANSI Z60 American Nurseryman and Landscape Association began developing this standard back in 1929 Became an ANSI document in 1949 Current version is 2004 . ANSI A300 The Tree Care Industry Association convened a consensus body .
Astrophysics needs input of practically all sub-disciplines of physics and thus a course on astrophysics cannot be self-contained. However, the course should be accessible to students with just a general introduction to physics. Few sections of the text that are somewhat more advanced and that can be omitted are marked by stars. I will be glad to receive feedback from the readers of these .
Aber es gibt natürlich auch Jungs, die beste Freunde sind, und Mädchen, die Cliquen haben. Oder Mädchen und Jungs, die befreundet sind. Doch nicht jeder findet sofort einen Freund oder eine Freundin. Dann steht man womöglich alleine herum und bekommt schnell das Gefühl, dass andere gegen einen sind. Zum Beispiel, wenn in Deiner Straße die Nachbarsmädchen über Dich lästern, weil Du ein .
7 BAB II TINJAUAN PUSTAKA 2.1 Deskripsi Tanaman Jambu Monyet (Anacardium occidentale L.)Jambu monyet berasal dari Brazil, tersebar di daerah tropik dan ditemukan pada ketinggian antara 1-1.200 mdpl.
As climate science and the Earth’s climate have continued to evolve over recent decades, increasing evidence of anthropogenic infl uences on climate change has been found. Correspondingly, the IPCC has made increasingly more defi nitive statements about human impacts on climate. Debate has stimulated a wide variety of climate change research.
the civil war in 1949, establishing the People's Republic of China (PRC), it became evident that the geopolitics of Asia had shifted against the colonial powers. It was also evident that the struggle over Asian independence would become a catalyst for the expansion of the cold war into Asia. For the United States, the “loss of China .
management, as well as corporate governance, in order to mitigate risks and build corporate compliance capacities. This Manual is designed so that its contents can be updated and revised annually. . RISK MANAGEMENT IS the identi cation, assessment, and prioritisation of risks
