New Functions from Old Functions

.3 New Functions rom Old Functions In this section we start with the basic unctions we discussed in Section. and obtain new unctions b shiting, stretching, and relecting their graphs. We also show how to combine pairs o unctions b the standard arithmetic operations and b composition. Transormations o Functions B appling certain transormations to the graph o a given unction we can obtain the graphs o certain related unctions. This will give us the abilit to sketch the graphs o man unctions quickl b hand. It will also enable us to write equations or given graphs. Let s irst consider translations. I c is a positive number, then the graph o c is just the graph o shited upward a distance o c units (because each -coordinate is increased b the same number c). Likewise, i t c, where c, then the value o t at is the same as the value o at c (c units to the let o ). Thereore, the graph o c is just the graph o shited c units to the right (see Figure ). Vertical and Horizontal Shits Suppose c. To obtain the graph o c, shit the graph o a distance c units upward c, shit the graph o a distance c units downward c, shit the graph o a distance c units to the right c, shit the graph o a distance c units to the let

SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 39 =ƒ+c =cƒ (c>) =(+c) c =ƒ =(-c) =(_) =ƒ c c = ƒ c c =ƒ-c =_ƒ FIGURE Translating the graph o ƒ FIGURE Stretching and relecting the graph o ƒ Now let s consider the stretching and relecting transormations. I c, then the graph o c is the graph o stretched b a actor o c in the vertical direction (because each -coordinate is multiplied b the same number c). The graph o is the graph o relected about the -ais because the point, is replaced b the point,. (See Figure and the ollowing chart, where the results o other stretching, compressing, and relecting transormations are also given.) Vertical and Horizontal Stretching and Relecting Suppose c. To obtain the graph o c, stretch the graph o verticall b a actor o c c, compress the graph o verticall b a actor o c c, compress the graph o horizontall b a actor o c c, stretch the graph o horizontall b a actor o c, relect the graph o about the -ais, relect the graph o about the -ais Figure 3 illustrates these stretching transormations when applied to the cosine unction with c. For instance, in order to get the graph o cos we multipl the -coordinate o each point on the graph o cos b. This means that the graph o cos gets stretched verticall b a actor o. = cos =cos = cos =cos FIGURE 3 =cos =cos

4 CHAPTER FUNCTIONS AND MODELS EXAMPLE Given the graph o s, use transormations to graph s, s, s, s, and s. SOLUTION The graph o the square root unction s, obtained rom Figure 3(a) in Section., is shown in Figure 4(a). In the other parts o the igure we sketch s b shiting units downward, s b shiting units to the right, s b relecting about the -ais, s b stretching verticall b a actor o, and s b relecting about the -ais. _ (a) =œ FIGURE 4 (b) =œ - EXAMPLE Sketch the graph o the unction () 6. SOLUTION (c) =œ - (d) =_œ (e) =œ () =œ _ Completing the square, we write the equation o the graph as This means we obtain the desired graph b starting with the parabola and shiting 3 units to the let and then unit upward (see Figure 5). 6 3 (_3, ) _3 _ FIGURE 5 (a) = (b) =(+3)@+ EXAMPLE 3 Sketch the graphs o the ollowing unctions. (a) sin (b) sin SOLUTION (a) We obtain the graph o sin rom that o sin b compressing horizontall b a actor o (see Figures 6 and 7). Thus, whereas the period o sin is, the period o sin is. =sin =sin π π π 4 π π FIGURE 6 FIGURE 7

SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 4 (b) To obtain the graph o sin, we again start with sin. We relect about the -ais to get the graph o sin and then we shit unit upward to get sin. (See Figure 8.) =-sin FIGURE 8 π π 3π π EXAMPLE 4 Figure 9 shows graphs o the number o hours o dalight as unctions o the time o the ear at several latitudes. Given that Ankara, Turke, is located at approimatel 4 N latitude, ind a unction that models the length o dalight at Ankara. 8 6 4 FIGURE 9 Graph o the length o dalight rom March through December at various latitudes Hours 8 6 4 6 N 5 N 4 N 3 N N Mar. Apr. Ma June Jul Aug. Sept. Oct. Nov. Dec. Source: Lucia C. Harrison, Dalight, Twilight, Darkness and Time (New York: Silver, Burdett, 935) page 4. SOLUTION Notice that each curve resembles a shited and stretched sine unction. B looking at the blue curve we see that, at the latitude o Ankara, dalight lasts about 4.8 hours on June and 9. hours on December, so the amplitude o the curve (the actor b which we have to stretch the sine curve verticall) is 4.8 9..8. B what actor do we need to stretch the sine curve horizontall i we measure the time t in das? Because there are about 365 das in a ear, the period o our model should be 365. But the period o sin t is, so the horizontal stretching actor is c 365. We also notice that the curve begins its ccle on March, the 8th da o the ear, so we have to shit the curve 8 units to the right. In addition, we shit it units upward. Thereore, we model the length o dalight in Ankara on the tth da o the ear b the unction L t.8 sin 365 t 8 Another transormation o some interest is taking the absolute value o a unction. I, then according to the deinition o absolute value, when

4 CHAPTER FUNCTIONS AND MODELS and when. This tells us how to get the graph o rom the graph o : The part o the graph that lies above the -ais remains the same; the part that lies below the -ais is relected about the -ais. EXAMPLE 5 Sketch the graph o the unction. SOLUTION We irst graph the parabola in Figure (a) b shiting the parabola downward unit. We see that the graph lies below the -ais when, so we relect that part o the graph about the -ais to obtain the graph o in Figure (b). FIGURE (a) = - (b) = - Combinations o Functions Two unctions and t can be combined to orm new unctions t, t, t, and t in a manner similar to the wa we add, subtract, multipl, and divide real numbers. I we deine the sum t b the equation t t then the right side o Equation makes sense i both and t are deined, that is, i belongs to the domain o and also to the domain o t. I the domain o is A and the domain o t is B, then the domain o t is the intersection o these domains, that is, A B. Notice that the sign on the let side o Equation stands or the operation o addition o unctions, but the sign on the right side o the equation stands or addition o the numbers and t. Similarl, we can deine the dierence t and the product t, and their domains are also A B. But in deining the quotient t we must remember not to divide b. Algebra o Functions Let and t be unctions with domains A and B. Then the unctions t, t, t, and t are deined as ollows: t t t t domain A B domain A B t t domain A B t domain A B t t

SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 43 Another wa to solve 4 : - + - _ EXAMPLE 6 I s and t s4, ind the unctions t, t, t, and t. SOLUTION The domain o s is,. The domain o t s4 consists o all numbers such that 4, that is, 4. Taking square roots o both sides, we get, or, so the domain o t is the interval,. The intersection o the domains o and t is,,, Thus, according to the deinitions, we have t s s4 t s s4 t s s4 s4 3 t s s4 4 Notice that the domain o t is the interval, ; we have to eclude because t. The graph o the unction t is obtained rom the graphs o and t b graphical addition. This means that we add corresponding -coordinates as in Figure. Figure shows the result o using this procedure to graph the unction t rom Eample 6. 5 =(+g)() =(+g)() 4 3 = g(a) (a)+g(a) =œ 4-.5 =ƒ (a) g(a).5 ƒ=œ a FIGURE FIGURE Composition o Functions There is another wa o combining two unctions to get a new unction. For eample, suppose that u su and u t. Since is a unction o u and u is, in turn, a unction o, it ollows that is ultimatel a unction o. We compute this b substitution: u t s

44 CHAPTER FUNCTIONS AND MODELS The procedure is called composition because the new unction is composed o the two given unctions and t. In general, given an two unctions and t, we start with a number in the domain o t and ind its image t. I this number t is in the domain o, then we can calculate the value o t. The result is a new unction h t obtained b substituting t into. It is called the composition (or composite) o and t and is denoted b t ( circle t ). Deinition Given two unctions and t, the composite unction t (also called the composition o and t) is deined b t t The domain o t is the set o all in the domain o t such that t is in the domain o. In other words, t is deined whenever both t and t are deined. The best wa to picture t is b either a machine diagram (Figure 3) or an arrow diagram (Figure 4). FIGURE 3 The g machine is composed o the g machine (irst) and then the machine. (input) g g() { } (output) g g FIGURE 4 Arrow diagram or g { } EXAMPLE 7 I and t 3, ind the composite unctions and t. SOLUTION We have t t t 3 3 t t t 3 NOTE You can see rom Eample 7 that, in general, t t. Remember, the notation t means that the unction t is applied irst and then is applied second. In Eample 7, t is the unction that irst subtracts 3 and then squares; t is the unction that irst squares and then subtracts 3. EXAMPLE 8 I s and t s, ind each unction and its domain. (a) t (b) t (c) (d) t t SOLUTION (a) t t (s ) ss s 4 The domain o t is,.

SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 45 (b) t t t(s) s s For s to be deined we must have. For s s to be deined we must have I a b, then a b. s, that is, s, or 4. Thus, we have 4, so the domain o t is the closed interval, 4. (c) (s) ss s 4 The domain o is,. (d) t t t t t(s ) s s This epression is deined when both and s. The irst inequalit means, and the second is equivalent to s, or 4, or. Thus,, so the domain o t t is the closed interval,. It is possible to take the composition o three or more unctions. For instance, the composite unction t h is ound b irst appling h, then t, and then as ollows: t h t h EXAMPLE 9 Find t h i, t, and h 3. SOLUTION t h t h t 3 3 3 3 So ar we have used composition to build complicated unctions rom simpler ones. But in calculus it is oten useul to be able to decompose a complicated unction into simpler ones, as in the ollowing eample. EXAMPLE Given F cos 9, ind unctions, t, and h such that F t h. SOLUTION Since F cos 9, the ormula or F sas: First add 9, then take the cosine o the result, and inall square. So we let h 9 t cos Then t h t h t 9 cos 9 cos 9 F.3 Eercises. Suppose the graph o is given. Write equations or the graphs that are obtained rom the graph o as ollows. (a) Shit 3 units upward. (b) Shit 3 units downward. (c) Shit 3 units to the right. (d) Shit 3 units to the let. (e) Relect about the -ais. () Relect about the -ais. (g) Stretch verticall b a actor o 3. (h) Shrink verticall b a actor o 3.

46 CHAPTER FUNCTIONS AND MODELS. Eplain how the ollowing graphs are obtained rom the graph o. (a) 5 (b) 5 (c) (d) 5 (e) 5 () 5 3 6. 3 3. The graph o is given. Match each equation with its graph and give reasons or our choices. (a) 4 (b) 3 (c) 3 (d) 4 (e) 6 @ 6! 7. 5 _4 3 # _.5 $ _6 _3 3 6 % _3 4. The graph o is given. Draw the graphs o the ollowing unctions. (a) 4 (b) 4 (c) (d) 3 8. (a) How is the graph o sin related to the graph o sin? Use our answer and Figure 6 to sketch the graph o sin. (b) How is the graph o s related to the graph o s? Use our answer and Figure 4(a) to sketch the graph o s. 9 4 Graph the unction b hand, not b plotting points, but b starting with the graph o one o the standard unctions given in Section., and then appling the appropriate transormations. 9. 3... 4 3 5. The graph o is given. Use it to graph the ollowing unctions. (a) (b) ( ) (c) (d) 3. cos 4. 4 sin 3 5. sin 7. s 3 8. 4 3 9. 8. s 3.. 4 tan 3. 4. sin 6. 4 4 6 7 The graph o s3 is given. Use transormations to create a unction whose graph is as shown..5 =œ 3-3 5. The cit o New Delhi, India, is located near latitude 3 N. Use Figure 9 to ind a unction that models the number o hours o dalight at New Delhi as a unction o the time o ear. To check the accurac o our model, use the act that on March 3 the Sun rises at 6:3 A.M. and sets at 6:39 P.M. in New Delhi. 6. A variable star is one whose brightness alternatel increases and decreases. For the most visible variable star, Delta Cephei, the time between periods o maimum brightness is 5.4 das, the average brightness (or magnitude) o the star

SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 47 is 4., and its brightness varies b.35 magnitude. Find a unction that models the brightness o Delta Cephei as a unction o time. 7. (a) How is the graph o ( ) related to the graph o? (b) Sketch the graph o sin. (c) Sketch the graph o. 8. Use the given graph o to sketch the graph o. Which eatures o are the most important in sketching? Eplain how the are used. s 36. 3, t 37. sin, t s 38. 3, t 5 3 39., t 4. s 3, t 4 4 Find t h. 4. s, t, h 3 4., t cos, h s 3 9 3 Use graphical addition to sketch the graph o t. 9. 43 46 Epress the unction in the orm t. 43. F 44. F sin(s) 45. u t scos t u t tan t tan t 46. g 47 49 Epress the unction in the orm t h. 47. H 3 48. 49. H sec 4 (s) H s 8 3. 5. Use the table to evaluate each epression. (a) t (b) t (c) (d) t t (e) t 3 () t 6 g 3 4 5 6 3 4 5 t 6 3 3 3 3 Find t, t, t, and t and state their domains. 3. 3, t 3 3. s, t s 5. Use the given graphs o and t to evaluate each epression, or eplain wh it is undeined. (a) t (b) t (c) t (d) t 6 (e) t t () 4 33 34 Use the graphs o and t and the method o graphical addition to sketch the graph o t. 33., t g 34. 3, t 35 4 Find the unctions (a) t, (b) t, (c), and (d) t t and their domains. 35., t

48 CHAPTER FUNCTIONS AND MODELS 5. Use the given graphs o and t to estimate the value o t or 5, 4, 3,..., 5. Use these estimates to sketch a rough graph o t. 53. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed o 6 cm s. (a) Epress the radius r o this circle as a unction o the time t (in seconds). (b) I A is the area o this circle as a unction o the radius, ind A r and interpret it. 54. A spherical balloon is being inlated and the radius o the balloon is increasing at a rate o cm s. (a) Epress the radius r o the balloon as a unction o the time t (in seconds). (b) I V is the volume o the balloon as a unction o the radius, ind V r and interpret it. 55. A ship is moving at a speed o 3 km h parallel to a straight shoreline. The ship is 6 km rom shore and it passes a lighthouse at noon. (a) Epress the distance s between the lighthouse and the ship as a unction o d, the distance the ship has traveled since noon; that is, ind so that s d. (b) Epress d as a unction o t, the time elapsed since noon; that is, ind t so that d t t. (c) Find t. What does this unction represent? 56. An airplane is ling at a speed o 35 km h at an altitude o one mile and passes directl over a radar station at time t. (a) Epress the horizontal distance d (in kilometres) that the plane has lown as a unction o t. (b) Epress the distance s between the plane and the radar station as a unction o d. (c) Use composition to epress s as a unction o t. 57. The Heaviside unction H is deined b H t i t i t It is used in the stud o electric circuits to represent the sudden surge o electric current, or voltage, when a switch is instantaneousl turned on. (a) Sketch the graph o the Heaviside unction. g (b) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t and volts are applied instantaneousl to the circuit. Write a ormula or V t in terms o H t. (c) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t 5 seconds and 4 volts are applied instantaneousl to the circuit. Write a ormula or V t in terms o H t. (Note that starting at t 5 corresponds to a translation.) 58. The Heaviside unction deined in Eercise 57 can also be used to deine the ramp unction cth t, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph o the ramp unction th t. (b) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t and the voltage is graduall increased to volts over a 6-second time interval. Write a ormula or V t in terms o H t or t 6. (c) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t 7 seconds and the voltage is graduall increased to volts over a period o 5 seconds. Write a ormula or V t in terms o H t or t 3. 59. Let and t be linear unctions with equations m b and t m b. Is t also a linear unction? I so, what is the slope o its graph? 6. I ou invest dollars at 4% interest compounded annuall, then the amount A o the investment ater one ear is A.4. Find A A, A A A, and A A A A. What do these compositions represent? Find a ormula or the composition o n copies o A. 6. (a) I t and h 4 4 7, ind a unction such that t h. (Think about what operations ou would have to perorm on the ormula or t to end up with the ormula or h.) (b) I 3 5 and h 3 3, ind a unction t such that t h. 6. I 4 and h 4, ind a unction t such that t h. 63. (a) Suppose and t are even unctions. What can ou sa about t and t? (b) What i and t are both odd? 64. Suppose is even and t is odd. What can ou sa about t? 65. Suppose t is an even unction and let h t. Is h alwas an even unction? 66. Suppose t is an odd unction and let h t. Is h alwas an odd unction? What i is odd? What i is even?