MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

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1 MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ] (, ] [ [, ) ( ) (, ) [ ) [, ) no solution ( ] (, ] [ ] [, ] ll rel numers (, ) You re responsile for knowing ll 3 wys ove to represent set of numers tht is n inequlity; with the inequlity symol, grphed on numer line, or in intervl nottion. 1

2 You my rememer from pst eperiences of grphing on numer line, tht you used closed dot on the numer line to indicte tht the numer itself ws included in the set. You proly used n open dot on the numer line to indicte tht the numer itself ws not included in the set. In this clss, we will use prenthesis to replce the open dot nd rcket to replce the closed dot. Students re epected to know how to represent n intervl on numer line s grph on numer line, using intervls nottion, or using the inequlity symols. Emple 0: () Represent the set of numers < 1, using numer line nd intervl nottion. () Represent the set of numers, using numer line nd intervl nottion. (c) Represent the set of numers 5 < 0, using numer line nd intervl nottion. (d) Represent the set of numers (3, ), using n inequlity symol nd using numer line. (e) Represent the set of numers (, 100], using n inequlity symol nd using numer line. (f) Represent the numers represented on the numer line elow in oth intervl nottion nd using n inequlity symol. [ ) -1 7

3 Definition: A reltion is ny set of ordered pirs. The set of first components in the ordered pirs is clled the domin of the reltion. The set of second components is clled the rnge of the reltion. Reltions my e represented s sets, tles, digrms, grphs, or equtions. Definition: A function is reltion in which no two ordered pirs hve the sme first components ut different second components. Ech element or component of the domin (input vlues) is pired to one nd only one element or component of the rnge (output vlues). Reltions or functions cn e represented y sets of ordered pirs, tles of ordered pirs, mppings of ordered pirs, grphs, or equtions (s seen in the net few emples). Emple 1: Which reltions elow represent functions? Stte the domins nd rnges. ) {(9,81), (4,16), (5,5), (,4), ( 6,36)} Function? Domin: Rnge: ) y -1 0 Function? Domin: Rnge: c) Function? DOMAIN RANGE d) Function? Domin: (1,-1) (3,-1) Rnge: (-3,-3) (continued on the net pge) 3

4 e) y Function? Domin: Rnge: f) y 5 3 Function? Domin: g) s 3r Function? Domin: Emple : Which elow represent functions? Wht is the domin? Wht is the rnge? h) {(,6), (7,8), (,0)} i) q() (continued on the net pge) 4

5 j) 3 4 DOMAIN 8 RANGE k) Function Nottion: Functions cn e nmed y using letters. This nme cn e used to write the function. For emple; the function h represented y y cn e written s h( ). This type of nottion is known s function nottion. The element of the domin, the input (the ) is inside the prentheses nd the result, the output, (the y or h()) is the mtching element of the rnge. Emple : Given the function g( ) 3 1, find the following function vlues. ) g( ) ) g(0) c) g(4) d) g( ) 5

6 Emple 3: Given the functions f ( ) nd F( ), find the following. 3 1 ) f (3 ) g) F(3 ) ) F( 1) h) f ( 1) c) f ( r ) i) F( r ) 1 1 d) F j) f e) f ( ) f () k) F( ) F() f ) f ( h) f ( ) 6

7 A function my e defined y more thn one eqution, such s the piecewise-defined function in emple 4. Emple 4: Given this piecewise-defined function, find the following. if 1 f( ) if 1 ) f(0) ) f(1) c) f ( ) if 1 d) f if 7

8 (lesson prt ) In most functions the implied domin is the set of ll rel numers (inputs) tht yield rel numer vlues for the function vlues (outputs). In rel life prolems, the domin nd rnge re only the vlues tht re resonle. Emple 5: Find the domin of ech function. ) f ( ) 3 d) g( ) ) G( ) e) h( ) c) d( t) 50 t, where d is the distnce in miles trveled y cr in t hours g) volume of sphere: V 4 3 r 3 8

9 f) A tennis ll is tossed verticlly upwrd from height of 5 feet ccording to the height function h(t) = 16t + 1t + 5, where h us the height of the tennis ll in feet nd t is time in seconds. When does the ll hit the ground? (Round to the nerest hundredth of second, if necessry.) Emple 6: A mnufcturer cn produce nd sell gdgets per week. The totl cost of producing these gdgets (in dollrs) is given y C( ) The revenue from the sle of these gdgets (in dollrs) is R( ) 6.6. () Find function to represent the profit from the production nd sell of gdgets per week. () Wht is the profit from the sle of 5 gdgets? Emple 7: Write cost function (cost s function of numer of songs) for this sitution. An Internet site where student cn downlod songs chrges $10 registrtion fee plus $0.75 per song. 9

10 Emple 8: The demnd function (price/cot) for type of cot if given y p D( n) n, where p is the price in dollrs nd n is the quntity demnded (numer of cots). ) Find the price for demnd of 55 cots. ) Find the quntity demnded for cot t the price of $3. Emple 9: Producing units of hmurger pckges costs C( ) The revenue is R( ) 16. (Both C nd R re in dollrs.) ) Write profit function P(). Find the rek-even quntity. ) Find the profit from 50 units of hmurger pckges. c) Find the numer of units tht must e produced for profit of $

11 Emple 10: Forensic scientists (CSI) use the lengths of certin ones to clculte the height of person. The following functions represent person s height sed on the length of the femur (r), the one from the knee to the hip socket. For women: h( r) r For men: h( r) r ) Find the height of mn with femur mesuring 59 cm. 6 cm. ) Find the height of womn with femur mesuring 48 cm. 55 cm. Emple 11: A compny tht mnufctures icycles hs fied cost of $100,000 month. It costs $100 on verge to produce ech icycle. The totl cost per month for the compny is the sum of its fied cost nd the vrile costs (costs of production of the icycles). Write cost function, C, s function of the numer of icycles produced,. Find C (90) nd C (00). Interpret ech. 11

12 Emple 1: A cr ws purchsed for $,500. The vlue of the cr decresed y $300 per yer for the first si yers. Write function tht descries the vlue of the cr, V, fter yers, where 0 6. Find nd interpret V (3) nd V(8). Emple 13: 0 if 4 p ( ) if 4 0 if 0 ) Find p( 4), p(0), nd p(3). ) Find p ( + ) if 4 < + < 0 c) Find p (5 3) if