Functions. Introduction

Transcription

1 Functions,00 P, y Fiure Standard and Poor s Inde with dividends reinvested (credit "bull": modiication o work by Prayitno Hadinata; credit "raph": modiication o work by MeasurinWorth) Chapter OUtline. Functions and Function Notation. Domain and Rane. Rates o Chane and Behavior o Graphs. Composition o Functions. Transormation o Functions. Absolute Value Functions.7 Inverse Functions Introduction Toward the end o the twentieth century, the values o stocks o internet and technoloy companies rose dramatically. As a result, the Standard and Poor s stock market averae rose as well. Fiure tracks the value o that initial investment o just under $00 over the 0 years. It shows that an investment that was worth less than $00 until about 99 skyrocketed up to about $,00 by the beinnin o 000. That ive-year period became known as the dot-com bubble because so many internet startups were ormed. As bubbles tend to do, thouh, the dot-com bubble eventually burst. Many companies rew too ast and then suddenly went out o business. The result caused the sharp decline represented on the raph beinnin at the end o 000. Notice, as we consider this eample, that there is a deinite relationship between the year and stock market averae. For any year we choose, we can determine the correspondin value o the stock market averae. In this chapter, we will eplore these kinds o relationships and their properties. This OpenSta book is available or ree at

2 SECTION. Composition o Functions Learnin Objectives In this section, you will: Combine unctions usin alebraic operations. Create a new unction by composition o unctions. Evaluate composite unctions. Find the domain o a composite unction. Decompose a composite unction into its component unctions.. Composition o Functions Suppose we want to calculate how much it costs to heat a house on a particular day o the year. The cost to heat a house will depend on the averae daily temperature, and in turn, the averae daily temperature depends on the particular day o the year. Notice how we have just deined two relationships: The cost depends on the temperature, and the temperature depends on the day. Usin descriptive variables, we can notate these two unctions. The unction C(T) ives the cost C o heatin a house or a iven averae daily temperature in T derees Celsius. The unction T(d) ives the averae daily temperature on day d o the year. For any iven day, Cost = C(T(d)) means that the cost depends on the temperature, which in turns depends on the day o the year. Thus, we can evaluate the cost unction at the temperature T(d). For eample, we could evaluate T() to determine the averae daily temperature on the th day o the year. Then, we could evaluate the cost unction at that temperature. We would write C(T()). Cost or the temperature C(T()) Temperature on day By combinin these two relationships into one unction, we have perormed unction composition, which is the ocus o this section. Combinin Functions Usin Alebraic Operations Function composition is only one way to combine eistin unctions. Another way is to carry out the usual alebraic operations on unctions, such as addition, subtraction, multiplication and division. We do this by perormin the operations with the unction outputs, deinin the result as the output o our new unction. Suppose we need to add two columns o numbers that represent a husband and wie s separate annual incomes over a period o years, with the result bein their total household income. We want to do this or every year, addin only that year s incomes and then collectin all the data in a new column. I w(y) is the wie s income and h(y) is the husband s income in year y, and we want T to represent the total income, then we can deine a new unction. T(y) = h(y) + w(y) I this holds true or every year, then we can ocus on the relation between the unctions without reerence to a year and write T = h + w Just as or this sum o two unctions, we can deine dierence, product, and ratio unctions or any pair o unctions that have the same kinds o inputs (not necessarily numbers) and also the same kinds o outputs (which do have to be numbers so that the usual operations o alebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think o addin, subtractin, multiplyin, and dividin unctions. This OpenSta book is available or ree at

3 CHAPTER Functions For two unctions () and () with real number outputs, we deine new unctions +,,, and by the relations ( + )() = () + () ( )() = () () ()() = ()() () = () () Eample Perormin Alebraic Operations on Functions Find and simpliy the unctions ( )() and (), iven () = and () =. Are they the same unction? Solution Bein by writin the eneral orm, and then substitute the iven unctions. No, the unctions are not the same. ( )() = () () ( )() = ( ) ( )() = ( )() = ( ) () = () () () = where () = ( + )( ) where () = + where Note: For (), the condition is necessary because when =, the denominator is equal to 0, which makes the unction undeined. Try It # Find and simpliy the unctions ()() and ( )(). Are they the same unction? Create a Function by Composition o Functions () = and () = Perormin alebraic operations on unctions combines them into a new unction, but we can also create unctions by composin unctions. When we wanted to compute a heatin cost rom a day o the year, we created a new unction that takes a day as input and yields a cost as output. The process o combinin unctions so that the output o one unction becomes the input o another is known as a composition o unctions. The resultin unction is known as a composite unction. We represent this combination by the ollowin notation: ( )() = (())

4 SECTION. Composition o Functions We read the let-hand side as composed with at, and the riht-hand side as o o. The two sides o the equation have the same mathematical meanin and are equal. The open circle symbol is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the unctions themselves without reerrin to any particular input value. Composition is a binary operation that takes two unctions and orms a new unction, much as addition or multiplication takes two numbers and ives a new number. However, it is important not to conuse unction composition with multiplication because, as we learned above, in most cases (()) ()(). It is also important to understand the order o operations in evaluatin a composite unction. We ollow the usual convention with parentheses by startin with the innermost parentheses irst, and then workin to the outside. In the equation above, the unction takes the input irst and yields an output (). Then the unction takes () as an input and yields an output (()). (), the output o is the input o ( )() = (()) is the input o In eneral, and are dierent unctions. In other words, in many cases ( ()) ( ()) or all. We will also see that sometimes two unctions can be composed only in one speciic order. For eample, i () = and () = +, then (()) = ( + ) = ( + ) but = + + ( ()) = ( ) = + These epressions are not equal or all values o, so the two unctions are not equal. It is irrelevant that the epressions happen to be equal or the sinle input value =. Note that the rane o the inside unction (the irst unction to be evaluated) needs to be within the domain o the outside unction. Less ormally, the composition has to make sense in terms o inputs and outputs. composition o unctions When the output o one unction is used as the input o another, we call the entire operation a composition o unctions. For any input and unctions and, this action deines a composite unction, which we write as such that ( )() = (()) The domain o the composite unction is all such that is in the domain o and () is in the domain o. It is important to realize that the product o unctions is not the same as the unction composition (()), because, in eneral, ()() (()). Eample Determinin whether Composition o Functions is Commutative Usin the unctions provided, ind (()) and ( ()). Determine whether the composition o the unctions is commutative. () = + () = This OpenSta book is available or ree at

5 CHAPTER Functions Solution Let s bein by substitutin () into (). Now we can substitute () into (). (()) = ( ) + = + = 7 ( ()) = ( + ) = = + We ind that ( ()) (()), so the operation o unction composition is not commutative. Eample Interpretin Composite Functions The unction c(s) ives the number o calories burned completin s sit-ups, and s(t) ives the number o sit-ups a person can complete in t minutes. Interpret c(s()). Solution The inside epression in the composition is s(). Because the input to the s-unction is time, t = represents minutes, and s() is the number o sit-ups completed in minutes. Usin s() as the input to the unction c(s) ives us the number o calories burned durin the number o sit-ups that can be completed in minutes, or simply the number o calories burned in minutes (by doin sit-ups). Eample Investiatin the Order o Function Composition Suppose () ives miles that can be driven in hours and (y) ives the allons o as used in drivin y miles. Which o these epressions is meaninul: ((y)) or ( ())? Solution The unction y = () is a unction whose output is the number o miles driven correspondin to the number o hours driven. number o miles = (number o hours) The unction (y) is a unction whose output is the number o allons used correspondin to the number o miles driven. This means: number o allons = (number o miles) The epression (y) takes miles as the input and a number o allons as the output. The unction () requires a number o hours as the input. Tryin to input a number o allons does not make sense. The epression ((y)) is meaninless. The epression () takes hours as input and a number o miles driven as the output. The unction (y) requires a number o miles as the input. Usin () (miles driven) as an input value or (y), where allons o as depends on miles driven, does make sense. The epression ( ()) makes sense, and will yield the number o allons o as used,, drivin a certain number o miles, (), in hours. Q & A Are there any situations where ((y)) and ( ()) would both be meaninul or useul epressions? Yes. For many pure mathematical unctions, both compositions make sense, even thouh they usually produce dierent new unctions. In real-world problems, unctions whose inputs and outputs have the same units also may ive compositions that are meaninul in either order. Try It # The ravitational orce on a planet a distance r rom the sun is iven by the unction G(r). The acceleration o a planet subjected to any orce F is iven by the unction a(f). Form a meaninul composition o these two unctions, and eplain what it means.

6 SECTION. Composition o Functions Evaluatin Composite Functions Once we compose a new unction rom two eistin unctions, we need to be able to evaluate it or any input in its domain. We will do this with speciic numerical inputs or unctions epressed as tables, raphs, and ormulas and with variables as inputs to unctions epressed as ormulas. In each case, we evaluate the inner unction usin the startin input and then use the inner unction s output as the input or the outer unction. Evaluatin Composite Functions Usin Tables When workin with unctions iven as tables, we read input and output values rom the table entries and always work rom the inside to the outside. We evaluate the inside unction irst and then use the output o the inside unction as the input to the outside unction. Eample Usin a Table to Evaluate a Composite Function Usin Table, evaluate (()) and ( ()). () () 8 7 Table Solution To evaluate (()), we start rom the inside with the input value. We then evaluate the inside epression () usin the table that deines the unction : () =. We can then use that result as the input to the unction, so () is replaced by and we et (). Then, usin the table that deines the unction, we ind that () = 8. () = (()) = () = 8 To evaluate ( ()), we irst evaluate the inside epression () usin the irst table: () =. Then, usin the table or, we can evaluate ( ()) = () = Table shows the composite unctions and as tables. () (()) () ( ()) 8 Table Try It # Usin Table, evaluate (()) and ( ()). Evaluatin Composite Functions Usin Graphs When we are iven individual unctions as raphs, the procedure or evaluatin composite unctions is similar to the process we use or evaluatin tables. We read the input and output values, but this time, rom the - and y-aes o the raphs. This OpenSta book is available or ree at

7 CHAPTER Functions How To Given a composite unction and raphs o its individual unctions, evaluate it usin the inormation provided by the raphs.. Locate the iven input to the inner unction on the -ais o its raph.. Read o the output o the inner unction rom the y-ais o its raph.. Locate the inner unction output on the -ais o the raph o the outer unction.. Read the output o the outer unction rom the y-ais o its raph. This is the output o the composite unction. Eample Usin a Graph to Evaluate a Composite Function Usin Fiure, evaluate (()). () () 7 7 (a) 7 7 (b) Fiure Solution To evaluate (()), we start with the inside evaluation. See Fiure. () 7 (, ) 7 () 7 (, ) 7 () = () = Fiure We evaluate () usin the raph o (), indin the input o on the -ais and indin the output value o the raph at that input. Here, () =. We use this value as the input to the unction. (()) = () We can then evaluate the composite unction by lookin to the raph o (), indin the input o on the -ais and readin the output value o the raph at this input. Here, () =, so ( ()) =.

8 SECTION. Composition o Functions 7 Analysis Fiure shows how we can mark the raphs with arrows to trace the path rom the input value to the output value. () () Fiure Try It # Usin Fiure, evaluate ( ()). Evaluatin Composite Functions Usin Formulas When evaluatin a composite unction where we have either created or been iven ormulas, the rule o workin rom the inside out remains the same. The input value to the outer unction will be the output o the inner unction, which may be a numerical value, a variable name, or a more complicated epression. While we can compose the unctions or each individual input value, it is sometimes helpul to ind a sinle ormula that will calculate the result o a composition ( ()). To do this, we will etend our idea o unction evaluation. Recall that, when we evaluate a unction like (t) = t t, we substitute the value inside the parentheses into the ormula wherever we see the input variable. How To Given a ormula or a composite unction, evaluate the unction.. Evaluate the inside unction usin the input value or variable provided.. Use the resultin output as the input to the outside unction. Eample 7 Evaluatin a Composition o Functions Epressed as Formulas with a Numerical Input Given (t) = t t and h() = +, evaluate (h()). Solution Because the inside epression is h(), we start by evaluatin h() at. Then (h()) = (), so we evaluate (t) at an input o. h() = () + h() = (h()) = () (h()) = (h()) = 0 Analysis It makes no dierence what the input variables t and were called in this problem because we evaluated or speciic numerical values. This OpenSta book is available or ree at

9 8 CHAPTER Functions Try It # Given (t) = t t and h() = +, evaluate a. h( ()) b. h( ( )) Findin the Domain o a Composite Function As we discussed previously, the domain o a composite unction such as is dependent on the domain o and the domain o. It is important to know when we can apply a composite unction and when we cannot, that is, to know the domain o a unction such as. Let us assume we know the domains o the unctions and separately. I we write the composite unction or an input as (()), we can see riht away that must be a member o the domain o in order or the epression to be meaninul, because otherwise we cannot complete the inner unction evaluation. However, we also see that () must be a member o the domain o, otherwise the second unction evaluation in (()) cannot be completed, and the epression is still undeined. Thus the domain o consists o only those inputs in the domain o that produce outputs rom belonin to the domain o. Note that the domain o composed with is the set o all such that is in the domain o and () is in the domain o. domain o a composite unction The domain o a composite unction (()) is the set o those inputs in the domain o or which () is in the domain o. How To Given a unction composition (()), determine its domain.. Find the domain o.. Find the domain o.. Find those inputs in the domain o or which () is in the domain o. That is, eclude those inputs rom the domain o or which () is not in the domain o. The resultin set is the domain o. Eample 8 Find the domain o Findin the Domain o a Composite Function ( )() where () = and () = Solution The domain o () consists o all real numbers ecept =, since that input value would cause us to divide by 0. Likewise, the domain o consists o all real numbers ecept. So we need to eclude rom the domain o () that value o or which () =. = = = = So the domain o is the set o all real numbers ecept and. This means that We can write this in interval notation as or,, (, )

10 SECTION. Composition o Functions 9 Findin the Domain o a Composite Function Involvin Radicals Find the domain o ( )() where () = + and () = Solution Because we cannot take the square root o a neative number, the domain o is (, ]. Now we check the domain o the composite unction ( )() = + For ( )() = +, + 0, since the radicand o a square root must be positive. Since the square roots are positive, 0, 0, which ives a domain o (, ]. Analysis This eample shows that knowlede o the rane o unctions (speciically the inner unction) can also be helpul in indin the domain o a composite unction. It also shows that the domain o can contain values that are not in the domain o, thouh they must be in the domain o. Try It # Find the domain o ( )() where () = and () = + Decomposin a Composite Function into its Component Functions In some cases, it is necessary to decompose a complicated unction. In other words, we can write it as a composition o two simpler unctions. There may be more than one way to decompose a composite unction, so we may choose the decomposition that appears to be most epedient. Eample 9 Decomposin a Function Write () = as the composition o two unctions. Solution We are lookin or two unctions, and h, so () = (h()). To do this, we look or a unction inside a unction in the ormula or (). As one possibility, we miht notice that the epression is the inside o the square root. We could then decompose the unction as We can check our answer by recomposin the unctions. Try It #7 h() = and () = (h()) = ( ) = Write () = as the composition o two unctions. + Access these online resources or additional instruction and practice with composite unctions. Composite Functions ( Composite Function Notation Application ( Composite Functions Usin Graphs ( Decompose Functions ( Composite Function Values ( This OpenSta book is available or ree at

11 0 CHAPTER Functions. section EXERCISES Verbal. How does one ind the domain o the quotient o two unctions,?. I the order is reversed when composin two unctions, can the result ever be the same as the answer in the oriinal order o the composition? I yes, ive an eample. I no, eplain why not.. What is the composition o two unctions,?. How do you ind the domain or the composition o two unctions,? Alebraic. Given () = + and () =, ind +,,, and. Determine the domain or each unction in interval notation. 7. Given () = + and () =, ind +,,, and. Determine the domain or each unction in interval notation.. Given () = + and () =, ind +,,, and. Determine the domain or each unction in interval notation. 8. Given () = and () =, ind +,,, and. Determine the domain or each unction in interval notation. 9. Given () = and () =, ind +,,, and. Determine the domain or each unction in interval notation.. Given () = + and () =, ind the ollowin: a. ( ()) b. ( ()) c. ( ()) d. ( )() e. ( )( ) 0. Given () = and () =, ind. Determine the domain or each unction in interval notation. For the ollowin eercises, use each pair o unctions to ind (()) and ( ()). Simpliy your answers.. () = +, () = +. () = +, () = +. () =, () = + +. () =, () =. () =, () = () =, () = + For the ollowin eercises, use each set o unctions to ind ((h())). Simpliy your answers. 8. () = +, () =, and h() = 9. () = +, () =, and h() = + 0. Given () =, and () =, ind the ollowin: a. ( )() b. the domain o ( )() in interval notation c. ( )() d. the domain o ( )() e.. Given () = and () =, ind the ollowin: a. ( )() b. the domain o ( )() in interval notation

12 SECTION. Section Eercises. Given the unctions () = and () = +, ind the ollowin: a. ( )() b. ( )(). Given unctions p() = and m() =, state the domain o each o the ollowin unctions usin interval notation: p() a. b. p(m()) c. m(p()) m(). Given unctions q() = and h() = 9, state the domain o each o the ollowin unctions usin interval notation. q() a. b. q(h()) c. h(q()) h(). For () = and () =, write the domain o ( )() in interval notation. For the ollowin eercises, ind unctions () and () so the iven unction can be epressed as h() = (()).. h() = ( + ) 7. h() = ( ) 8. h() = 9. h() = ( + ) 0. h() = +. h() =. h() = ( ). h() =. h() = h() = h() = +. h() = +. h() = ( ) 7. h() = ( ) 0. h() =. h() = + Graphical For the ollowin eercises, use the raphs o, shown in Fiure, and, shown in Fiure, to evaluate the epressions. () () 7 7 Fiure Fiure. ( ()). ( ()). ( ()). ( (0)). ( ()) 7. ( ()) 8. ( ()) 9. ( (0)) This OpenSta book is available or ree at

13 CHAPTER Functions For the ollowin eercises, use raphs o (), shown in Fiure, (), shown in Fiure 7, and h(), shown in Fiure 8, to evaluate the epressions. () () () () () h() Fiure Fiure 7 Fiure 8 0. ( ()). ( ()). ( ()). ( ()). (h()). h( ()). ( (h())) 7. ( ( ( ))) Numeric For the ollowin eercises, use the unction values or and shown in Table to evaluate each epression () () Table 8. ( (8)) 9. ( ()) 0. ( ()). ( ()). ( ()). ( ()). ( ()). ( ()) For the ollowin eercises, use the unction values or and shown in Table to evaluate the epressions. 0 () 9 7 () Table. ( )() 7. ( )() 8. ( )() 9. ( )() 70. ( )() 7. ( )() For the ollowin eercises, use each pair o unctions to ind ((0)) and ( (0)). 7. () = + 8, () = 7 7. () = + 7, () = 7. () = +, () = 7. () =, () = + + For the ollowin eercises, use the unctions () = + and () = + to evaluate or ind the composite unction as indicated. 7. ( ()) 77. ( ()) 78. ( ( )) 79. ( )()

14 SECTION. Section Eercises Etensions For the ollowin eercises, use () = + and () =. 80. Find ( )() and ( )(). Compare the two answers. 8. Find ( )() and ( )(). 8. What is the domain o ( )()? 8. What is the domain o ( )()? 8. Let () =. a. Find ( )(). b. Is ( )() or any unction the same result as the answer to part (a) or any unction? Eplain. For the ollowin eercises, let F () = ( + ), () =, and () = True or False: ( )() = F (). 8. True or False: ( )() = F (). For the ollowin eercises, ind the composition when () = + or all 0 and () =. 87. ( )() ; ( )() 88. ( )(a) ; ( )(a) 89. ( )() ; ( )() Real-World ApplICATIOns 90. The unction D(p) ives the number o items that will be demanded when the price is p. The production cost C() is the cost o producin items. To determine the cost o production when the price is $, you would do which o the ollowin? a. Evaluate D(C()). b. Evaluate C(D()). c. Solve D(C()) =. d. Solve C(D(p)) =. 9. A store oers customers a 0 % discount on the price o selected items. Then, the store takes o an additional % at the cash reister. Write a price unction P() that computes the inal price o the item in terms o the oriinal price. (Hint: Use unction composition to ind your answer.) 9. A orest ire leaves behind an area o rass burned in an epandin circular pattern. I the radius o the circle o burnin rass is increasin with time accordin to the ormula r(t) = t +, epress the area burned as a unction o time, t (minutes). 9. The unction A(d) ives the pain level on a scale o 0 to 0 eperienced by a patient with d millirams o a pain- reducin dru in her system. The millirams o the dru in the patient s system ater t minutes is modeled by m(t). Which o the ollowin would you do in order to determine when the patient will be at a pain level o? a. Evaluate A(m()). b. Evaluate m(a()). c. Solve A(m(t)) =. d. Solve m(a(d)) =. 9. A rain drop hittin a lake makes a circular ripple. I the radius, in inches, rows as a unction o time in minutes accordin to r(t) = t +, ind the area o the ripple as a unction o time. Find the area o the ripple at t =. 9. Use the unction you ound in the previous eercise to ind the total area burned ater minutes. 9. The radius r, in inches, o a spherical balloon is related to the volume, V, by r(v) = V. Air is π pumped into the balloon, so the volume ater t seconds is iven by V(t) = 0 + 0t. a. Find the composite unction r(v(t)). b. Find the eact time when the radius reaches 0 inches. This OpenSta book is available or ree at The number o bacteria in a rerierated ood product is iven by N(T) = T T +, < T <, where T is the temperature o the ood. When the ood is removed rom the rerierator, the temperature is iven by T(t) = t +., where t is the time in hours. a. Find the composite unction N(T(t)). b. Find the time (round to two decimal places) when the bacteria count reaches,7.