Section 1.6: Graphs of Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative

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1 Section.6: Graphs of Functions, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 0, Carl Stitz.

2 .6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time in that section looking at functions graphicall because the were, after all, just sets of points in the plane. Then in Section. we described a function as a process and defined the notation necessar to work with functions algebraicall. So now it s time to look at functions graphicall again, onl this time we ll do so with the notation defined in Section.. We start with what should not be a surprising connection. The Fundamental Graphing Principle for Functions The graph of a function f is the set of points which satisf the equation = f(). That is, the point (, ) is on the graph of f if and onl if = f(). Eample.6.. Graph f() = 6. Solution. To graph f, we graph the equation = f(). To this end, we use the techniques outlined in Section... Specificall, we check for intercepts, test for smmetr, and plot additional points as needed. To find the -intercepts, we set = 0. Since = f(), this means f() = 0. f() = 6 0 = 6 0 = ( )( + ) factor = 0 or + = 0 =, So we get (, 0) and (, 0) as -intercepts. To find the -intercept, we set = 0. Using function notation, this is the same as finding f(0) and f(0) = = 6. Thus the -intercept is (0, 6). As far as smmetr is concerned, we can tell from the intercepts that the graph possesses none of the three smmetries discussed thus far. (You should verif this.) We can make a table analogous to the ones we made in Section.., plot the points and connect the dots in a somewhat pleasing fashion to get the graph below on the right. f() (, f()) 6 (, 6) 0 (, 0) (, ) 0 6 (0, 6) 6 (, 6) (, ) 0 (, 0) 6 (, 6) 7 6 6

3 9 Relations and Functions Graphing piecewise-defined functions is a bit more of a challenge. Eample.6.. Graph: f() = { if <, if Solution. We proceed as before finding intercepts, testing for smmetr and then plotting additional points as needed. To find the -intercepts, as before, we set f() = 0. The twist is that we have two formulas for f(). For <, we use the formula f() =. Setting f() = 0 gives 0 =, so that = ±. However, of these two answers, onl = fits in the domain < for this piece. This means the onl -intercept for the < region of the -ais is (, 0). For, f() =. Setting f() = 0 gives 0 =, or =. Since = satisfies the inequalit, we get (, 0) as another -intercept. Net, we seek the -intercept. Notice that = 0 falls in the domain <. Thus f(0) = 0 = ields the -intercept (0, ). As far as smmetr is concerned, ou can check that the equation = is smmetric about the -ais; unfortunatel, this equation (and its smmetr) is valid onl for <. You can also verif = possesses none of the smmetries discussed in the Section... When plotting additional points, it is important to keep in mind the restrictions on for each piece of the function. The sticking point for this function is =, since this is where the equations change. When =, we use the formula f() =, so the point on the graph (, f()) is (, ). However, for all values less than, we use the formula f() =. As we have discussed earlier in Section., there is no real number which immediatel precedes = on the number line. Thus for the values = 0.9, = 0.99, = 0.999, and so on, we find the corresponding values using the formula f() =. Making a table as before, we see that as the values sneak up to = in this fashion, the f() values inch closer and closer to =. To indicate this graphicall, we use an open circle at the point (, ). Putting all of this information together and plotting additional points, we get f() (, f()) (0.9,.9) (0.99,.0) (0.999,.00) We ve just stepped into Calculus here!

4 .6 Graphs of Functions 9 In the previous two eamples, the -coordinates of the -intercepts of the graph of = f() were found b solving f() = 0. For this reason, the are called the zeros of f. Definition.9. The zeros of a function f are the solutions to the equation f() = 0. In other words, is a zero of f if and onl if (, 0) is an -intercept of the graph of = f(). Of the three smmetries discussed in Section.., onl two are of significance to functions: smmetr about the -ais and smmetr about the origin. Recall that we can test whether the graph of an equation is smmetric about the -ais b replacing with and checking to see if an equivalent equation results. If we are graphing the equation = f(), substituting for results in the equation = f( ). In order for this equation to be equivalent to the original equation = f() we need f( ) = f(). In a similar fashion, we recall that to test an equation s graph for smmetr about the origin, we replace and with and, respectivel. Doing this substitution in the equation = f() results in = f( ). Solving the latter equation for gives = f( ). In order for this equation to be equivalent to the original equation = f() we need f( ) = f(), or, equivalentl, f( ) = f(). These results are summarized below. Testing the Graph of a Function for Smmetr The graph of a function f is smmetric ˆ about the -ais if and onl if f( ) = f() for all in the domain of f. ˆ about the origin if and onl if f( ) = f() for all in the domain of f. For reasons which won t become clear until we stud polnomials, we call a function even if its graph is smmetric about the -ais or odd if its graph is smmetric about the origin. Apart from a ver specialized famil of functions which are both even and odd, functions fall into one of three distinct categories: even, odd, or neither even nor odd. Eample.6.. Determine analticall if the following functions are even, odd, or neither even nor odd. Verif our result with a graphing calculator.. f() =. g() =. h() =. i() = {. j() = p() = + if < 0 +, if 0 Solution. The first step in all of these problems is to replace with and simplif. Wh are we so dismissive about smmetr about the -ais for graphs of functions? An ideas?

5 96 Relations and Functions. f() = f( ) = f( ) = f( ) = f() ( ) Hence, f is even. The graphing calculator furnishes the following. This suggests that the graph of f is smmetric about the -ais, as epected.. g() = g( ) = ( ) ( ) g( ) = It doesn t appear that g( ) is equivalent to g(). To prove this, we check with an value. After some trial and error, we see that g() = whereas g( ) =. This proves that g is not even, but it doesn t rule out the possibilit that g is odd. (Wh not?) To check if g is odd, we compare g( ) with g() Hence, g is odd. Graphicall, Suggests is about the etent of what it can do. g() = = g() = g( )

6 .6 Graphs of Functions 97 The calculator indicates the graph of g is smmetric about the origin, as epected.. h() = h( ) = ( ) ( ) h( ) = + Once again, h( ) doesn t appear to be equivalent to h(). We check with an value, for eample, h() = but h( ) =. This proves that h is not even and it also shows h is not odd. (Wh?) Graphicall, The graph of h appears to be neither smmetric about the -ais nor the origin.. i() = i( ) = i( ) = ( ) ( ) ( ) + The epression i( ) doesn t appear to be equivalent to i(). However, after checking some values, for eample = ields i() = and i( ) =, it appears that i( ) does, in fact, equal i(). However, while this suggests i is even, it doesn t prove it. (It does, however, prove

7 98 Relations and Functions i is not odd.) To prove i( ) = i(), we need to manipulate our epressions for i() and i( ) and show that the are equivalent. A clue as to how to proceed is in the numerators: in the formula for i(), the numerator is and in i( ) the numerator is. To re-write i() with a numerator of, we need to multipl its numerator b. To keep the value of the fraction the same, we need to multipl the denominator b as well. Thus i() = = = ( ) ( ) ( ) + Hence, i() = i( ), so i is even. The calculator supports our conclusion.. j() = 00 j( ) = ( ) 00 j( ) = + 00 The epression for j( ) doesn t seem to be equivalent to j(), so we check using = to get j() = 00 and j( ) = 00. This rules out j being even. However, it doesn t rule out j being odd. Eamining j() gives j() = 00 ( j() = ) 00 j() = The epression j() doesn t seem to match j( ) either. Testing = gives j() = 9 0 and j( ) = 0, so j is not odd, either. The calculator gives:

8 .6 Graphs of Functions 99 The calculator suggests that the graph of j is smmetric about the -ais which would impl that j is even. However, we have proven that is not the case. 6. Testing the graph of = p() for smmetr is complicated b the fact p() is a piecewisedefined function. As alwas, we handle this b checking the condition for smmetr b checking it on each piece of the domain. We first consider the case when < 0 and set about finding the correct epression for p( ). Even though p() = + for < 0, p( ) + here. The reason for this is that since < 0, > 0 which means to find p( ), we need to use the other formula for p(), namel p() = +. Hence, for < 0, p( ) = ( )+ = + = p(). For 0, p() = + and we have two cases. If > 0, then < 0 so p( ) = ( ) + = + = p(). If = 0, then p(0) = = p( 0). Hence, in all cases, p( ) = p(), so p is even. Since p(0) = but p( 0) = p(0) =, we also have p is not odd. While graphing = p() is not onerous to do b hand, it is instructive to see how to enter this into our calculator. B using some of the logical commands, we have: The calculator bears shows that the graph appears to be smmetric about the -ais. There are two lessons to be learned from the last eample. The first is that sampling function values at particular values is not enough to prove that a function is even or odd despite the fact that j( ) = j(), j turned out not to be odd. Secondl, while the calculator ma suggest mathematical truths, it is the Algebra which proves mathematical truths. 6 Consult our owner s manual, instructor, or favorite video site! 6 Or, in other words, don t rel too heavil on the machine!

9 00 Relations and Functions.6. General Function Behavior The last topic we wish to address in this section is general function behavior. As ou shall see in the net several chapters, each famil of functions has its own unique attributes and we will stud them all in great detail. The purpose of this section s discussion, then, is to la the foundation for that further stud b investigating aspects of function behavior which appl to all functions. To start, we will eamine the concepts of increasing, decreasing and constant. Before defining the concepts algebraicall, it is instructive to first look at them graphicall. Consider the graph of the function f below. (,.) 7 6 (6,.) 6 7 (, ) (, 6) 6 7 (, 6) 8 9 (, 8) The graph of = f() Reading from left to right, the graph starts at the point (, ) and ends at the point (6,.). If we imagine walking from left to right on the graph, between (, ) and (,.), we are walking uphill ; then between (,.) and (, 8), we are walking downhill ; and between (, 8) and (, 6), we are walking uphill once more. From (, 6) to (, 6), we level off, and then resume walking uphill from (, 6) to (6,.). In other words, for the values between and (inclusive), the -coordinates on the graph are getting larger, or increasing, as we move from left to right. Since = f(), the values on the graph are the function values, and we sa that the function f is increasing on the interval [, ]. Analogousl, we sa that f is decreasing on the interval [, ] increasing once more on the interval [, ], constant on [, ], and finall increasing once again on [, 6]. It is etremel important to notice that the behavior (increasing, decreasing or constant) occurs on an interval on the -ais. When we sa that the function f is increasing

10 .6 Graphs of Functions 0 on [, ] we do not mention the actual values that f attains along the wa. Thus, we report where the behavior occurs, not to what etent the behavior occurs. 7 Also notice that we do not sa that a function is increasing, decreasing or constant at a single value. In fact, we would run into serious trouble in our previous eample if we tried to do so because = is contained in an interval on which f was increasing and one on which it is decreasing. (There s more on this issue and man others in the Eercises.) We re now read for the more formal algebraic definitions of what it means for a function to be increasing, decreasing or constant. Definition.0. Suppose f is a function defined on an interval I. We sa f is: ˆ increasing on I if and onl if f(a) < f(b) for all real numbers a, b in I with a < b. ˆ decreasing on I if and onl if f(a) > f(b) for all real numbers a, b in I with a < b. ˆ constant on I if and onl if f(a) = f(b) for all real numbers a, b in I. It is worth taking some time to see that the algebraic descriptions of increasing, decreasing and constant as stated in Definition.0 agree with our graphical descriptions given earlier. You should look back through the eamples and eercise sets in previous sections where graphs were given to see if ou can determine the intervals on which the functions are increasing, decreasing or constant. Can ou find an eample of a function for which none of the concepts in Definition.0 appl? Now let s turn our attention to a few of the points on the graph. Clearl the point (,.) does not have the largest value of all of the points on the graph of f indeed that honor goes to (6,.) but (,.) should get some sort of consolation prize for being the top of the hill between = and =. We sa that the function f has a local maimum 8 at the point (,.), because the -coordinate. is the largest -value (hence, function value) on the curve near 9 =. Similarl, we sa that the function f has a local minimum 0 at the point (, 8), since the -coordinate 8 is the smallest function value near =. Although it is tempting to sa that local etrema occur when the function changes from increasing to decreasing or vice versa, it is not a precise enough wa to define the concepts for the needs of Calculus. At the risk of being pedantic, we will present the traditional definitions and thoroughl vet the pathologies the induce in the Eercises. We have one last observation to make before we proceed to the algebraic definitions and look at a fairl tame, et helpful, eample. If we look at the entire graph, we see that the largest value (the largest function value) is. at = 6. In this case, we sa the maimum of f is.; similarl, the minimum of f is 8. 7 The notions of how quickl or how slowl a function increases or decreases are eplored in Calculus. 8 Also called relative maimum. 9 We will make this more precise in a moment. 0 Also called a relative minimum. Maima is the plural of maimum and mimima is the plural of minimum. Etrema is the plural of etremum which combines maimum and minimum. Sometimes called the absolute or global maimum. Again, absolute or global minimum can be used.

11 0 Relations and Functions We formalize these concepts in the following definitions. Definition.. Suppose f is a function with f(a) = b. ˆ ˆ We sa f has a local maimum at the point (a, b) if and onl if there is an open interval I containing a for which f(a) f() for all in I. The value f(a) = b is called a local maimum value of f in this case. We sa f has a local minimum at the point (a, b) if and onl if there is an open interval I containing a for which f(a) f() for all in I. The value f(a) = b is called a local minimum value of f in this case. ˆ The value b is called the maimum of f if b f() for all in the domain of f. ˆ The value b is called the minimum of f if b f() for all in the domain of f. It s important to note that not ever function will have all of these features. Indeed, it is possible to have a function with no local or absolute etrema at all! (An ideas of what such a function s graph would have to look like?) We shall see eamples of functions in the Eercises which have one or two, but not all, of these features, some that have instances of each tpe of etremum and some functions that seem to def common sense. In all cases, though, we shall adhere to the algebraic definitions above as we eplore the wonderful diversit of graphs that functions provide us. Here is the tame eample which was promised earlier. It summarizes all of the concepts presented in this section as well as some from previous sections so ou should spend some time thinking deepl about it before proceeding to the Eercises. Eample.6.. Given the graph of = f() below, answer all of the following questions. (0, ) (, 0) (, 0) (, ) (, )

12 .6 Graphs of Functions 0. Find the domain of f.. Find the range of f.. List the -intercepts, if an eist.. List the -intercepts, if an eist.. Find the zeros of f. 6. Solve f() < Determine f(). 8. Solve f() =. 9. Find the number of solutions to f() =. 0. Does f appear to be even, odd, or neither?. List the intervals on which f is increasing.. List the intervals on which f is decreasing.. List the local maimums, if an eist.. List the local minimums, if an eist.. Find the maimum, if it eists. 6. Find the minimum, if it eists. Solution.. To find the domain of f, we proceed as in Section.. B projecting the graph to the -ais, we see that the portion of the -ais which corresponds to a point on the graph is everthing from to, inclusive. Hence, the domain is [, ].. To find the range, we project the graph to the -ais. We see that the values from to, inclusive, constitute the range of f. Hence, our answer is [, ].. The -intercepts are the points on the graph with -coordinate 0, namel (, 0) and (, 0).. The -intercept is the point on the graph with -coordinate 0, namel (0, ).. The zeros of f are the -coordinates of the -intercepts of the graph of = f() which are =,. 6. To solve f() < 0, we look for the values of the points on the graph where the -coordinate is less than 0. Graphicall, we are looking for where the graph is below the -ais. This happens for the values from to and again from to. So our answer is [, ) (, ]. 7. Since the graph of f is the graph of the equation = f(), f() is the -coordinate of the point which corresponds to =. Since the point (, 0) is on the graph, we have f() = To solve f() =, we look where = f() =. We find two points with a -coordinate of, namel (, ) and (, ). Hence, the solutions to f() = are = ±. 9. As in the previous problem, to solve f() =, we look for points on the graph where the -coordinate is. Even though these points aren t specified, we see that the curve has two points with a value of, as seen in the graph below. That means there are two solutions to f() =.

13 0 Relations and Functions 0. The graph appears to be smmetric about the -ais. This suggests that f is even.. As we move from left to right, the graph rises from (, ) to (0, ). This means f is increasing on the interval [, 0]. (Remember, the answer here is an interval on the -ais.). As we move from left to right, the graph falls from (0, ) to (, ). This means f is decreasing on the interval [0, ]. (Remember, the answer here is an interval on the -ais.). The function has its onl local maimum at (0, ) so f(0) = is the local minimum value.. There are no local minimums. Wh don t (, ) and (, ) count? Let s consider the point (, ) for a moment. Recall that, in the definition of local minimum, there needs to be an open interval I which contains = such that f( ) < f() for all in I different from. But if we put an open interval around = a portion of that interval will lie outside of the domain of f. Because we are unable to fulfill the requirements of the definition for a local minimum, we cannot claim that f has one at (, ). The point (, ) fails for the same reason no open interval around = stas within the domain of f.. The maimum value of f is the largest -coordinate which is. 6. The minimum value of f is the smallest -coordinate which is. With few eceptions, we will not develop techniques in College Algebra which allow us to determine the intervals on which a function is increasing, decreasing or constant or to find the local maimums and local minimums analticall; this is the business of Calculus. When we have need to find such beasts, we will resort to the calculator. Most graphing calculators have Minimum and Maimum features which can be used to approimate these values, as we now demonstrate. but does not prove Although, truth be told, there is onl one step of Calculus involved, followed b several pages of algebra.

14 .6 Graphs of Functions 0 Eample.6.. Let f() =. Use a graphing calculator to approimate the intervals on + which f is increasing and those on which it is decreasing. Approimate all etrema. Solution. Entering this function into the calculator gives Using the Minimum and Maimum features, we get To two decimal places, f appears to have its onl local minimum at (.7,.) and its onl local maimum at (.7,.). Given the smmetr about the origin suggested b the graph, the relation between these points shouldn t be too surprising. The function appears to be increasing on [.7,.7] and decreasing on (,.7] [.7, ). This makes. the (absolute) minimum and. the (absolute) maimum. Eample.6.6. Find the points on the graph of = ( ) which are closest to the origin. Round our answers to two decimal places. Solution. Suppose a point (, ) is on the graph of = ( ). Its distance to the origin (0, 0) is given b d = ( 0) + ( 0) = + = + [( ) ] Since = ( ) = + ( ) Given a value for, the formula d = + ( ) is the distance from (0, 0) to the point (, ) on the curve = ( ). What we have defined, then, is a function d() which we wish to

15 06 Relations and Functions minimize over all values of. To accomplish this task analticall would require Calculus so as we ve mentioned before, we can use a graphing calculator to find an approimate solution. Using the calculator, we enter the function d() as shown below and graph. Using the Minimum feature, we see above on the right that the (absolute) minimum occurs near =. Rounding to two decimal places, we get that the minimum distance occurs when =.00. To find the value on the parabola associated with =.00, we substitute.00 into the equation to get = ( ) = (.00 ) =.00. So, our final answer is (.00,.00). 6 (What does the value listed on the calculator screen mean in this problem?) 6 It seems sill to list a final answer as (.00,.00). Indeed, Calculus confirms that the eact answer to this problem is, in fact, (, ). As ou are well aware b now, the authors are overl pedantic, and as such, use the decimal places to remind the reader that an result garnered from a calculator in this fashion is an approimation, and should be treated as such.

16 .6 Graphs of Functions Eercises In Eercises -, sketch the graph of the given function. identif an intercepts and test for smmetr.. f() =. f() = State the domain of the function,. f() = +. f() =. f() = 6. f() = 7. f() = ( )( + ) 8. f() = 9. f() = 0. f() = +. f() =. f() = + In Eercises - 0, sketch the graph of the given piecewise-defined function. { { if. f() =. f() = if 0 if > if > 0 if < 0. f() = if 0 if > { if < 0 7. f() = if 0 if 9. f() = if < < if if 6. f() = if < < if { + if < 8. f() = if if 6 < < 0. f() = if < < if < < 9 In Eercises -, determine analticall if the following functions are even, odd or neither.. f() = 7. f() = 7 +. f() = 7. f() =. f() = 6. f() = 6 7. f() = 8. f() = + 9. f() = f() = f() =. f() =. f() = 0. f() =. f() =

17 08 Relations and Functions 6. f() = 7. f() = + 8. f() = + 9. f() = 0. f() = 9. f() = + In Eercises - 7, use the graph of = f() given below to answer the question.. Find the domain of f.. Find the range of f.. Determine f( ).. Solve f() =. 6. List the -intercepts, if an eist. 7. List the -intercepts, if an eist. 8. Find the zeros of f. 9. Solve f() Find the number of solutions to f() =.. Does f appear to be even, odd, or neither?. List the intervals where f is increasing.. List the intervals where f is decreasing.. List the local maimums, if an eist.. List the local minimums, if an eist. 6. Find the maimum, if it eists. 7. Find the minimum, if it eists.

18 .6 Graphs of Functions 09 In Eercises 8-7, use the graph of = f() given below to answer the question. 8. Find the domain of f. 9. Find the range of f. 60. Determine f(). 6. Solve f() =. 6. List the -intercepts, if an eist. 6. List the -intercepts, if an eist. 6. Find the zeros of f. 6. Solve f() Find the number of solutions to f() =. 67. Does f appear to be even, odd, or neither? 68. List the intervals where f is increasing. 69. List the intervals where f is decreasing. 70. List the local maimums, if an eist. 7. List the local minimums, if an eist. 7. Find the maimum, if it eists. 7. Find the minimum, if it eists. In Eercises 7-77, use our graphing calculator to approimate the local and absolute etrema of the given function. Approimate the intervals on which the function is increasing and those on which it is decreasing. Round our answers to two decimal places. 7. f() = f() = / ( ) 76. f() = f() = 9

19 0 Relations and Functions In Eercises 78-8, use the graphs of = f() and = g() below to find the function value. = f() = g() 78. (f + g)(0) 79. (f + g)() 80. (f g)() 8. (g f)() 8. (fg)() 8. (fg)() 8. ( f g ) () 8. ( g f ) () The graph below represents the height h of a Sasquatch (in feet) as a function of its age N in ears. Use it to answer the questions in Eercises N = h(n) 86. Find and interpret h(0). 87. How tall is the Sasquatch when she is ears old? 88. Solve h(n) = 6 and interpret. 89. List the interval over which h is constant and interpret our answer. 90. List the interval over which h is decreasing and interpret our answer.

20 .6 Graphs of Functions For Eercises 9-9, let f() = be the greatest integer function as defined in Eercise 7 in Section.. 9. Graph = f(). Be careful to correctl describe the behavior of the graph near the integers. 9. Is f even, odd, or neither? Eplain. 9. Discuss with our classmates which points on the graph are local minimums, local maimums or both. Is f ever increasing? Decreasing? Constant? In Eercises 9-9, use our graphing calculator to show that the given function does not have an etrema, neither local nor absolute. 9. f() = + 9. f() = In Eercise 7 in Section., we saw that the population of Sasquatch in Portage Count could be modeled b the function P (t) = 0t, where t = 0 represents the ear 80. Use t + our graphing calculator to analze the general function behavior of P. Will there ever be a time when 00 Sasquatch roam Portage Count? 97. Suppose f and g are both even functions. What can be said about the functions f + g, f g, fg and f g? What if f and g are both odd? What if f is even but g is odd? 98. One of the most important aspects of the Cartesian Coordinate Plane is its abilit to put Algebra into geometric terms and Geometr into algebraic terms. We ve spent most of this chapter looking at this ver phenomenon and now ou should spend some time with our classmates reviewing what we ve done. What major results do we have that tie Algebra and Geometr together? What concepts from Geometr have we not et described algebraicall? What topics from Intermediate Algebra have we not et discussed geometricall? It s now time to thoroughl vet the pathologies induced b the precise definitions of local maimum and local minimum. We ll do this b providing ou and our classmates a series of Eercises to discuss. You will need to refer back to Definition.0 (Increasing, Decreasing and Constant) and Definition. (Maimum and Minimum) during the discussion. 99. Consider the graph of the function f given below.

21 Relations and Functions (a) Show that f has a local maimum but not a local minimum at the point (, ). (b) Show that f has a local minimum but not a local maimum at the point (, ). (c) Show that f has a local maimum AND a local minimum at the point (0, ). (d) Show that f is constant on the interval [, ] and thus has both a local maimum AND a local minimum at ever point (, f()) where < <. 00. Using Eample.6. as a guide, show that the function g whose graph is given below does not have a local maimum at (, ) nor does it have a local minimum at (, ). Find its etrema, both local and absolute. What s unique about the point (0, ) on this graph? Also find the intervals on which g is increasing and those on which g is decreasing. 0. We said earlier in the section that it is not good enough to sa local etrema eist where a function changes from increasing to decreasing or vice versa. As a previous eercise showed, we could have local etrema when a function is constant so now we need to eamine some functions whose graphs do indeed change direction. Consider the functions graphed below. Notice that all four of them change direction at an open circle on the graph. Eamine each for local etrema. What is the effect of placing the dot on the -ais above or below the open circle? What could ou sa if no function value were assigned to = 0? (a) Function I (b) Function II

22 .6 Graphs of Functions (c) Function III (d) Function IV

23 Relations and Functions.6. Answers. f() = Domain: (, ) -intercept: (, 0) -intercept: (0, ) No smmetr. f() = Domain: (, ) -intercept: (, 0) -intercept: ( 0, ) No smmetr. f() = + Domain: (, ) -intercept: None -intercept: (0, ) Even. f() = Domain: (, ) -intercepts: (, 0), (, 0) -intercept: (0, ) Even. f() = Domain: (, ) -intercept: None -intercept: (0, ) Even

24 .6 Graphs of Functions 6. f() = Domain: (, ) -intercept: (0, 0) -intercept: (0, 0) Odd f() = ( )( + ) Domain: (, ) -intercepts: (, 0), (0, 0), (, 0) -intercept: (0, 0) No smmetr 8. f() = Domain: [, ) -intercept: (, 0) -intercept: None No smmetr f() = Domain: (, ] -intercept: (, 0) -intercept: (0, ) No smmetr

25 6 Relations and Functions 0. f() = + Domain: [, ) -intercept: (, 0) -intercept: (0, ) No smmetr. f() = Domain: (, ) -intercept: (0, 0) -intercept: (0, 0) Odd f() = + Domain: (, ) -intercept: None -intercept: (0, ) Even

26 .6 Graphs of Functions odd. neither. even. even. even 6. neither 7. odd 8. odd 9. even 0. neither. neither. even. even and odd. odd. even 6. even 7. neither 8. odd 9. odd 0. even. even. [, ]. [, ]. f( ) =. = 6. (, 0), (, 0), (, 0) 7. (0, ) 8.,, 9. [, ] [, ] 0.

27 8 Relations and Functions. neither. [, ], [0, ]. [, 0], [, ]. f( ) =, f() =. f(0) = 6. f( ) = 7. f( ) = 8. [, ] 9. [, ) 60. f() = 6. = 6. (, 0), (0, 0), (, 0) 6. (0, 0) 6., 0, 6. [, 0] {} neither 68. [, ) 69. [, ], (, ] 70. none 7. f( ) =, f() = 7. none 7. f( ) = 7. No absolute maimum Absolute minimum f(.) 7.6 Local minimum at (.8, 9.) Local maimum at (0.,.7) Local minimum at (., 7.6) Increasing on [.8, 0.], [., ) Decreasing on (,.8], [0.,.] 76. Absolute maimum f(0) = Absolute minimum f(±) = 0 Local maimum at (0, ) No local minimum Increasing on [, 0] Decreasing on [0, ] 7. No absolute maimum No absolute minimum Local maimum at (0, 0) Local minimum at (.60,.8) Increasing on (, 0], [.60, ) Decreasing on [0,.60] 77. Absolute maimum f(.).0 Absolute minimum f(.).0 Local maimum (.,.0) Local minimum (.,.0) Increasing on [.,.] Decreasing on [,.], [., ] 78. (f + g)(0) = 79. (f + g)() = 80. (f g)() = 8. (g f)() = 0 ( ) ( ) 8. (fg)() = 9 8. (fg)() = 6 8. f g () = 0 8. g f () = 86. h(0) =, so the Sasquatch is feet tall at birth. 87. h() = 6, so the Saquatch is 6 feet tall when she is ears old. 88. h(n) = 6 when N = and N = 60. This means the Sasquatch is 6 feet tall when she is and 60 ears old. 89. h is constant on [0, ]. This means the Sasquatch s height is constant (at 8 feet) for these ears.

28 .6 Graphs of Functions h is decreasing on [, 60]. This means the Sasquatch is getting shorter from the age of to the age of 60. (Sasquatchteoporosis, perhaps?) Note that f(.) =, but f(.) =, so f is neither even nor odd The graph of f() =.