On Range and Reflecting Functions About the Line y = mx
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1 On Range and Reflecting Functions About the Line = m Scott J. Beslin Brian K. Heck Jerem J. Becnel Dept.of Mathematics and Dept. of Mathematics and Dept. of Mathematics and Computer Science Computer Science Statistics Nicholls State Universit Nicholls State Universit Stephen F. Austin State Universit Thibodau, LA 7030 Thibodau, LA 7030 Nacogdoches, TX 7596 scott.beslin@nicholls.edu brian.heck@nicholls.edu becneljj@sfasu.edu Abstract: The authors eplore the importance of range and its relationship to continuousl differentiable functions that have inverses when their graphs are reflected about lines other than =. Some open questions are posed for the reader. Mathematics Subject Classification: 6A09 Ke Words: reflection, matri representation, range, inverse
2 On Range and Reflecting Functions About the Line = m I. Introduction In college algebra and beond, students stud functions and their graphs. In almost ever eercise, the are asked to specif the domain and range of a function. Most students have accepted the importance of the domain, given their work in solving equations involving roots and fractions, and later in addressing the intricacies of graphing and finding limits. However, man students avoid finding the range of the function; it is for them a more difficult process and man do not see the point in it. Depending on the level of the class, the instructor has several elementar avenues for illustrating the importance of range. We briefl mention a few. (i) Solving Equations The fact that = and sin = have no solutions is clearl related to the ranges of the functions involved. (ii) Finding Domains of Composite Functions Since the domain of the composite function f o g is the set { Domain( g) and g( ) Domain( f )}, the range of the function g is particularl important in determining the domain of this composite function. For eample, if g : g( ) =, then the domain of the composite go g is empt. (iii) Functions restricted in domain so that their inverses eist + is given b The trigonometric functions provide numerous eamples of such functions. The fact that 3π 3π Arcsin(sin 4 ) 4 is due to the fact that 3 π 4 is not in the range of the Arcsin function. In [], the authors generalize a problem from elementar calculus. In that paper, a new tpe of derivative is introduced as a method of solving pursuit problems. One of the principal
3 results concerning the solvabilit of such problems is directl related to the range of the Newton- Raphson function encountered in root-finding analsis. In this paper, we eplore a generalization related to inverse function. The ke result of this investigation illustrates the importance of range in a situation different from those cited above. II. Reflections and Reflection Matrices In an introductor algebra class, inverse functions are studied. Usuall, in addition to the algebraic method for computing the inverse of a one-to-one function, a graphical method is also eplored. Namel, if a one-to-one function is reflected about the line =, the result is its inverse. Obviousl, if the function is not one-to-one, it does not have an inverse, for the result of such a reflection is not even a function. But what if the reflection takes place about a different line? For eample, the function = is not one-to-one, and if reflected about =, we get the graph of = which is not a function (see Figure ). Figure. The graph of =
4 However, if we reflect = about the line =, we do in fact get a function, namel = (see Figure ). To see this informall, observe that = reflected about the - ais ields the function =. For reflection about the line =, a line parallel to the - ais, note that (0,0), the verte of =, clearl maps to (0, ). Figure. The graph of = For a better eample, consider the function = e. A reflection about = ields the natural logarithm function. However, we can also reflect it about the lines = and = and ield functions (see Figures 3 and 4 respectivel). We will further address the notion of reflecting functions in Section III. 3
5 The function Figure 3. = e reflected about = The function Figure 4. = e reflected about = 4
6 Recall that b the reflection of a point P = (, ) about the line = m, we mean the point P = (, ) such that the given line is the perpendicular bisector of the segment PP (see Figure 5). Figure 5. The definition of reflection about a line If P is on the line = m, then P = P. Intuitivel, then, if f is an function graphed with wet ink in the plane, and we fold the plane along the line about this line is the imprint of f after folding. Suppose that the line L defined b = m, the reflection of f = m makes an angle θ with the positive -ais, where m = tanθ. One wa of effecting the reflection of P = (, ) about L is given b the following procedure from linear algebra: 5
7 Motion () Rotate clockwise through angle θ () Reflect about the -ais (3) Rotate counterclockwise through θ Matri of Motion cosθ sinθ A = sinθ cosθ B 0 = 0 cosθ sinθ C = sinθ cosθ Thus the final reflection matri is R θ cos θ sin θ sinθ cosθ cos θ sin θ = A B C = = sinθ cosθ sin θ cos θ sin θ cos θ Using the appropriate trigonometric identities with m = tanθ, we have m m Rθ = R( m) = m + m m Note that m lim = ± m + m m and lim = 0 m ± m + so that R π 0 = R( ± ) = 0 is the usual matri for reflection about the -ais. In addition, is the matri of reflection about the -ais, and R 0 0 = R(0) = 0 R π = R() = 0 = 0 6
8 is the matri associated with inversion (reflection about = ). The following list of elementar linear algebraic properties of the matri R = R( m) is not difficult to verif. Some, like number (), are easil seen geometricall. We leave these properties as student eercises. () The determinant of R = R =. () R is invertible and R = R. (3) trace ( R ) = 0. (4) R is smmetric; that is R T = R. (5) R is an orthogonal matri. (6) The minimal polnomial of R is t ; its eigenvalues are ±. Clearl the product of two R -matrices is not necessaril an R -matri. For eample, 0 R(0) R() = 0 which is not smmetric. However, geometricall, if we reflect the product about the -ais once more, we obtain or equivalentl, 0 R(0) R() R(0) = = R( ) 0 R R = R R = R R (0) () ( ) (0) ( ) (0). Algebra and trigonometr can be used to show that this relationship holds more generall. If m = tanα and n = tan β represent slopes of lines through the origin, then 7
9 tanα tan β tan( α β ) = + tanα tan β m n = + mn π as long as the two lines are not perpendicular, in which case α β = and mn =. From (somewhat tedious) algebra, letting J 0 = R(0) = 0, we have ( ) m n R( m) R( n) J = R if mn. If the lines are perpendicular, then R( m) R( n) J must + mn effect a simple reflection about the -ais; hence in that case 0 R( m) R( n) J = = R( ± ) 0. Since J = J, we can sum up b writing for all real numbers m and n, m n ( + mn ) ( ) R J if mn, R( m) R( n) = R ± J if mn =. Taking appropriate limits, we ma etend this result to include the cases: ( ) R( m) R( ± ) = R J; m ( ) R( ± ) R( n) = R J; 0 R( ± ) R( ± ) =. 0 n If the set M is defined b M = { R( m) m is a real number or m = + or m = }, we ma define an operation * on M via: A* B = A B J. In the language of abstract algebra, the operation * is a binar operator on M which is neither commutative nor associative. 8
10 III. When Does Reflection Give a Function? We now eplore conditions on m such that the reflection of a function f b R( m ) ields another function. Geometricall, it is eas to see that the reflection of a linear function f ( ) = a + b b R( m ) is almost alwas another linear function. (This reflection will result in a vertical line if m a a = ± +.) But generall speaking, if we abandon graphing as a proof that the reflection is or is not a function, it is not algebraicall evident what the case ma be. For eample, reflecting the function f ( ) = e about the line = results in the parametric representation (for < t < + ): t e t (3 4 ) = 5 + and = (4 3 t ) 5 t e. 3 4 To see this, observe that R( ) = 5 4 3, so that 3 4 t 5 t = 4 3 e. From the parametric equations alone, it is not immediatel clear whether can be written as a function of. In what follows, we let f : be a continuousl differentiable ( C ) function, and ask: When is the reflection of f b R( m ) a function? The idea of range plas a significant part in the answer to this question. Reflection of f b R( m ) ields: m m t = m + m m f ( t), or ( t) = ( m ) t + mf ( t) m + and ( t) = mt + ( m ) f ( t) m +. 9
11 If ( t) 0, then ( t) > 0 for all t or ( t) < 0 for all t, b the continuit of and the Intermediate Value Theorem. In an case, function of. Observe that, eists and t =. So t = ( ) = o is a d d dt m m f t = = d m mf t d dt + ( ) ( ) + ( ) ; in particular, ( t) = 0 if m f ( t) =. m Result follows: Result : Let f : be a C function. If m is such that m Range( ) m f, then reflection of f b R( m ) ields a function. Note that reflection of a function b R (0) alwas ields a function. Corollar : If f : is a C non-decreasing function (with f 0 ), and m m 0 <, then reflection of f b R( m ) ields a function. + Eample: We return to the increasing function f ( ) = e. Since f ( ) = e, Range( f ) =. Solving < ields m < or 0 < m <. For instance, reflection of f about the line = m m 0 (eamined previousl) is a function. Corollar : If f : is a C non-increasing function (with 0 f ), and either < m < 0 or m >, then reflection of f b R( m ) ields a function. Suppose the reflection of f b R( m ) is a function = ( ). Since (, ) = ( ( t), ( t)) as before, a similar argument to the one preceding Result shows that if ( t) 0, then is a function of. Since is also a function of, we see that must be a one-to-one function of. Observing that ( t) = 0 whenever m f ( t) =, we state Result. m 0
12 Result : Let f : be a C function. For ever m such that m m Range( f ) and Range( f ). m m then reflection of f b R( m ) is a one-to-one function. Corollar 3: If f is increasing, then its reflection b R( m ) is a one-to-one function precisel when m {0,, }. In other words, the onl possible one-to-one reflections are f, f, and ( f ), corresponding to ( ) R m acting on f for m = 0,, and, respectivel. [For eample, reflection of f ( ) = e about = ields a function, but not a one-to-one function.] Eample: Let f ( ) = Arctan( ). Then f ( ) =, and so Range( f ) = (0,]. Solving + m m (0,] together with (0,] m m gives that reflection of f b R( m ) is a one-to-one function whenever m [, ] [,0] [ +,] [ +, ). The algebra involved here becomes less burdensome if one notices that reciprocals. IV. Concluding Remarks m m and m m are negative Results for reflections about lines of the form = m + b, b 0, ma be obtained from the foregoing theor b vertical or horizontal translations of the plane. Finall, we state a pair of open questions for investigation. Let f : be a C function which is invertible across the line = m in the sense of Section III. Let f m be the reflection of f about this line.. When is f o f = f o f? m m
13 d. How is ( f m ( ) d ) related to f ( )? Reference [] S. Beslin and D. Bane, A Different Tpe of Differentiabilit, PRIMUS, March 000 (Vol. X, No. ), pp Scott Beslin is currentl department head of Mathematics and Computer Science at Nicholls State Universit in Thibodau, Louisiana. Although his administrative duties are often demanding, he still enjos working with students and colleagues on mathematical research and eposition. Scott and the mathematics facult are presentl developing the Master of Science in Communit/Technical College Mathematics for online deliver Brian K. Heck received his Ph. D. in mathematics from Louisiana State Universit in 997. He began teaching at Nicholls State Universit in 998 and is currentl a tenured associate professor. Jerem J. Becnel was born in the small communit of Kraemer, Louisiana on August, 979. He finished his undergraduate studies in mathematics and computer science at Nicholls State Universit in Ma 00. In August 00 he went to Louisiana State Universit to pursue graduate studies in mathematics. He earned a Master of Science degree in mathematics from Louisiana State Universit in December 00 and received his doctorate from Louisiana State Universit in August 006. Jerem is currentl an assistant professor at Stephen F. Austin State Universit in Nacogdoches, Teas.
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