2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes

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1 Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion and he uni sep funcion. Imporan properies and definiions are saed. This secion can be used as a reference when he need arises. There are, of course, oher ypes of funcion which arise in engineering applicaions, such as rigonomeric, exponenial and logarihm funcions. These ohers are deal wih in Workbooks 4 o 6. Prerequisies Before saring his Secion you should... Learning Oucomes Afer compleing his Secion you should be able o... undersand wha is mean by a funcion and use funcional noaion 2 be able o plo graphs of funcions undersand wha is mean by a polynomial funcion, and a raional funcion be able o use and graph he modulus funcion be able o use and graph he uni sep funcion

2 . Polynomial Funcions A very imporan ype of funcion is he polynomial. Polynomial funcions are made up of muliples of non-negaive whole number powers of a variable, such as 3x 2, 7x 3 and so on. You are already familiar wih many such funcions. Oher examples include: P 0 () =6 P () =3 +9 (The linear funcion you have already me). P 2 (x) =3x 2 x +2 P 4 (z) =7z 4 + z 2 where, x and z are independen variables. Noe ha fracional and negaive powers of he independen variable are no allowed so ha f(x) =x and g(x) =x 3/2 are no polynomials. The funcion P 0 () =6is a polynomial - we can regard i as 6 0. By convenion a polynomial is wrien wih he powers eiher increasing or decreasing. For example he polynomial 3x +9x 2 x 3 +2 would be wrien as In general we have he following definiion: x 3 +9x 2 +3x +2 or 2+3x +9x 2 x 3 A polynomial expression has he form Key Poin a n x n + a n x n + a n 2 x n a 2 x 2 + a x + a 0 where n is a non-negaive ineger, a n, a n,...,a, a 0 are consans and x is a variable. A polynomial funcion P (x) has he form P (x) =a n x n + a n x n + a n 2 x n a 2 x 2 + a x + a 0 The degree of a polynomial or polynomial funcion is he value of he highes power. Referring o he examples lised above, polynomial P 2 has degree 2, because he erm wih he highes power is 3x 2, P 4 has degree 4, P has degree and P 0 has degree 0. Polynomials wih low degrees have special names given in Table. HELM (VERSION : March 8, 2004): Workbook Level 0 2

3 Table degree name a 0 consan ax + b linear ax 2 + bx + c 2 quadraic ax 3 + bx 2 + cx + d 3 cubic ax 4 + bx 3 + cx 2 + dx + e 4 quaric Typical graphs of some polynomial funcions are shown in Figure. In paricular, observe ha he graphs of he linear polynomials, P and Q 2 are sraigh lines. P 2 (x) =x 2 +3 Q (x) = x P (x) =2x P 3 (x) =x x 5 5 x Q 2 (x) = x 2 +2x 5 Q 3 (x) = x 3 +7x 6 Figure. Graphs of some ypical linear, quadraic and cubic polynomials Which of he polynomial graphs in Figure are odd and which are even? Are any periodic? Your soluion P 2 is even, P 3 is odd. None are peiodic Sae which of he following are polynomial funcions. For hose ha are, give he degree and name. a) f(x) =6x 2 +7x 3 2x 4 b) f() = c) g(x) = x x d) f(x) =6 e)g(x) = 6 Your soluion a) polynomial of degree 4 (quaric), b) polynomial of degree 3 (cubic), c) no a polynomial, d) polynomial of degree 0 (consan), e) polynomial of degree 0 (consan) 3 HELM (VERSION : March 8, 2004): Workbook Level 0

4 Exercises. Wrie down a polynomial of degree 3 wih independen variable. 2. Wrie down a funcion which is no a polynomial. 3. Explain why y =+x + x /2 is no a polynomial. 4. Sae he degree of he following polynomials: a) P () = 4 +7,b)P () = 3 +3,c)P () =, d) P () = 5. Wrie down a polynomial of degree 0 wih independen variable z. 6. Referring o Figure, sae which funcions are one-o-one and which are many-o-one. Answers. For example f() = For example y = x. 3. A erm such as x /2, wih a fracional index, is no allowed in a polynomial. 4. a) 4, b) 3, c) 0, d). 5. P (z) =3, for example. 6. P, Q and P 3 are one-o-one. The res are many-o-one. 2. Raional Funcions A raional funcion is formed by dividing one polynomial by anoher. Examples include R (x) = x +6 x 2 +, R 2() = , R 3(z) = 2z2 + z z 2 + z 2 For convenience we have labelled hese raional funcions R, R 2 and R 3. Key Poin A raional funcion has he form R(x) = P (x) Q(x) where P and Q are polynomial funcions, P is called he numeraor and Q is called he denominaor. The graphs of raional funcions can ake a variey of differen forms and can be difficul o plo by hand. Use of a graphics calculaor or compuer sofware can help. If you have access o a ploing package or calculaor i would be useful o obain graphs of hese funcions for yourself. The nex hree Examples allow you o explore some of he feaures of he graphs. HELM (VERSION : March 8, 2004): Workbook Level 0 4

5 Sudy carefully he graph in he following figure and he algebraic form of he raional funcion R (x) = x+2 and ry o answer he following quesions. x 2 + y 2 2 x Graph of R (x) = x+2 x 2 + a) For wha values of x, ifany, ishe denominaor zero? b) For wha values of x, if any, is he denominaor negaive? c) For wha values of x is he funcion negaive? d) Wha is he value of he funcion when x is zero? e) Wha happens o he funcion if x ges very large (say 0, 00...)? Subsiue some values o see. Your soluion Soluion. a) x 2 +is never zero, b) x 2 +is never negaive, c) only when x is less han 2, d) 2, e) R approaches zero because he x 2 erm in he denominaor becomes very large. Noe ha for large x values he graph ges closer and closer o he x axis. We say ha he x axis is a horizonal asympoe of his graph. Asking and answering quesions such as (a) o (c) above will help you o skech graphs of raional funcions. 5 HELM (VERSION : March 8, 2004): Workbook Level 0

6 Sudy he graph and he algebraic form of he funcion R 2 () = carefully and ry o answer he following quesions. The following figure shows is graph (he solid curve) Graph of R 2 () = a) Wha is he funcion value when =? b) Wha is he value of he denominaor when = 3/2? c) Wha do you hink happens o he graph of he funcion when = 3/2? Your soluion a) 0, b) 0 c) The funcion value ends o infiniy, he graph becomes infinie. Noe from pars b) and c) ha we mus exclude he value = 3/2 from he domain of his funcion because division by zero is no defined. A his poin as you can see he graph shoos off owards very large posiive values (we say i ends o posiive infiniy) if he poin is approached from he lef, and owards very large negaive values (we say i end o negaive infiniy) if he poin is approached from he righ. The doed line in he graph of R 2 (x) has equaion = 3. 2 I is approached by he curve as approaches 3 and is known as a verical asympoe. 2 HELM (VERSION : March 8, 2004): Workbook Level 0 6

7 Sudy he graph and he algebraic form of he funcion R 3 (z) = 2z2 +z (z )(z+2) carefully and ry o answer he following quesions. The graph of R 3 (z) is shown in he following figure z 0 Graph of R 3 (z) = 2z2 +z (z )(z+2) a) Wha is happening o he graph when z = 2 and when z =? b) Which values should be excluded from he domain of his funcion? c) Try subsiuing some large values for z (e.g. 0,00...). Wha happens o R 3 as z ges large? d) Is here a horizonal asympoe? e) Wha is he name given o he verical lines z =and z = 2? Your soluion a) denominaor is zero, R 3 ends o infiniy, b) z = 2 and z =, c) R 3 approaches he value 2, d) y = 2is a horizonal asympoe, e) verical asympoes Examples 2-4 are inended o give you some guidance so ha you will be able o skech raional funcions of your own. Each funcion mus be looked a individually bu some general guidelines are given below: Find he value of he funcion when he independen variable is zero. This is generally easy o evaluae and gives you a poin on he graph. Find values of he independen variable which make he denominaor zero. These values mus be excluded from he domain of he funcion and give rise o verical asympoes. 7 HELM (VERSION : March 8, 2004): Workbook Level 0

8 Find values of he independen variable which make he dependen variable zero. This gives you poins where he graph cus he horizonal axis (if a all). Sudy he behaviour of he funcion when x is large and posiive and when i is large and negaive. Are here any verical or horizonal asympoes? (Oblique asympoes may also occur bu hese are beyond he scope of he reamen given here). I is paricularly imporan for engineers o find values of he independen variable for which he denominaor is zero. These values are are known as he poles of he raional funcion. Sae he poles of he following raional funcions: a) f() = 3 +7 s+7 b) F (s) = (s+3)(s 3) c) r(x) = 2x+5 (x+)(x+2) d) f(x) = x. x 2 Your soluion In each case we locae he poles by seeking values of he independen variable which make he denominaor zero. a) 7, b) 3 or 3, c) or 2, d) x = In each case he calculaed values are he poles of he raional funcion. If you have access o a ploing package, plo hese funcions now. Exercises. Explain wha is mean by a raional funcion. 2. Sae he degree of he numeraor and he degree of he denominaor of he raional funcion R(x) = 3x2 +x+. x 3. Explain he erm pole of a raional funcion. 4. Referring o he graphs of R (x),r 2 () and R 3 (z) (in his secion), sae which funcions, if any, are one-o-one and which are many-o-one. 5. Wihou using a graphical calculaor plo graphs of y = and y =. Commen upon x x 2 wheher hese graphs are odd, even or neiher, wheher hey are coninuous or disconinuous, and sae he posiion of any poles. HELM (VERSION : March 8, 2004): Workbook Level 0 8

9 Answers. R(x) = P (x)/q(x) where P and Q are polynomials. 2. 2,, respecively. 3. The pole is a value of he independen variable which makes he denominaor zero. 4. All are many-o-one. 5. is odd, and disconinuous. Pole a x =0. is even and disconinuous. Pole a x =0. x x 2 3. The modulus funcion The modulus of a number is he size of ha number wih no regard paid o is sign. For example he modulus of 7 is7. The modulus of +7 is also 7. We can wrie his concisely using he modulus sign.sowecan wrie 7 =7and +7 =7. The modulus funcion is defined as follows: The modulus funcion is defined as f(x) = x = Key Poin { x x 0 x x < 0 The oupu from his funcion is simply he modulus of he inpu. A graph of his funcion is shown in he following figure. f(x) = x Graph of he modulus funcion x x 9 HELM (VERSION : March 8, 2004): Workbook Level 0

10 Draw up a able of values of he funcion f(x) = x 2 for values of x beween 3 and 5. Skech a graph of his funcion. Your soluion The able has been sared. Complee i for yourself. x f(x) Some poins on he graph are shown in he figure. Plo your calculaed poins on he graph. f(x) = x x Exercises. Skech a graph of he following funcions: a) f(x) =3 x. b) f(x) = x +. c) f(x) =7 x Is he modulus funcion one-o-one or many o one? Answers 2. Many-o-one 4. The uni sep funcion The uni sep funcion is defined as follows: Key Poin The uni sep funcion u(): u() = { 0 0 <0 HELM (VERSION : March 8, 2004): Workbook Level 0 0

11 Sudy his definiion carefully. You will see ha i is defined in wo pars, wih one expression o be used when is greaer han or equal o 0, and anoher expression o be used when is less han 0. The graph of his funcion is shown in he following figure. Noe ha he par of u() for which <0 lies on he -axis bu, for clariy, is shown as a disinc dashed line. u() Graph of he uni sep funcion There is a jump, or disconinuiy in he graph when =0. Tha is why we need o define he funcion in wo pars; one par for when is negaive, and one par for when is non-negaive. The poin wih coordinaes (0,) is par of he funcion defined on 0. The posiion of he disconinuiy may be shifed o he lef or righ. The graph of u( d) is shown in he nex figure. u( d) d Graph of u( d). In he previous wo figures he funcion akes he value 0 or. We can adjus he value by muliplying he funcion by any oher number we choose. The graph of 2u( 3) is shown in he nex figure. 2u( 3) 2 3 Graph of 2u( 3). Exercises. Skech graphs of he following funcions: a) u(), b) u(), c) u( ), d) u( + ), e) u( 3) u( 2), f) 3u(), g) 2u( 3). HELM (VERSION : March 8, 2004): Workbook Level 0

12 Answers. (a) (b) u() u() - (c) (d) u( ) u( +) - (e) (f) 3u() u( 3) u( 2) (g) u( 3) HELM (VERSION : March 8, 2004): Workbook Level 0 2