Inverse Functions Attachment It is essential to understand the dierence etween an inverse operation and an inverse unction. In the simplest terms, inverse operations in mathematics are two operations that have the opposite eect, such as addition and sutraction or operations that undo each other. Addition and sutraction are inverse operations, as are multiplication and division. This can e conused with an inverse element (or inverse operation). The inverse element, is deined or speciic sets o numers in relationship to a inary operation.. The ormal deinition o a group (o which one condition is that an inverse element exists) as it applies to set theory or inary operations ollows: A group (G, *) is a set G with a inary operation * that satisies the ollowing our axioms: Closure : For all a, in G, the result o a * is also in G. Associativity: For all a, and c in G, (a * ) * c = a * ( * c). Identity element: There exists an element e in G such that or all a in G, e * a = a * e = a. Inverse element: For each a in G, there exists an element in G such that a * = * a = e, where e is an identity element. For example, or the operation o addition or integers, the identity element is 0 (since we have a + 0 = 0 + a = a or all a ε W) and the inverse o a is a ecause a + ( a) = 0. This is also called the additive inverse. The multiplicative inverse (or the reciprocal) o a is 1 a. Using a mapping diagram, the inverse o shown is 1 (where 1 is not an exponent). 1 a d d a c e e c Formally, i is a unction with domain X, then 1 is its inverse unction i and only i or every 1 1 x X we have: ( x) ( x) = x =. We also get x = 1 (y) or y = (x). I a unction has an inverse then is said to e invertile. I an inverse exists, it is unique. The superscript " 1" is not an exponent. We have these inverse unctions in secondary school mathematics: y = x + a and y = x a, y = ax and y = x a, y = x and y = x, y = sinx and y = sin 1 x, y = x and y = log x, y = e x and y = lnx, y = ( x) and y = x dx (no longer in our curriculum). An inverse unction can e determined in three ways: i) using a set or list o elements, ii) using a graph, or iii) using an equation. ( )
For a set o ordered pair elements, the inverse relation is otained y interchanging the order o the elements in each pair. For example, i we have set A = {(1, 5), (, 7), (3, 11)} then its inverse A 1 is {(5, 1), (7, ), (11, 3)} The inverse o a relation deined y a graph is ound y relecting the graph in the line y = x. An example is shown elow. In this case, the inverse is not a unction. In general, the inverse o a unction is a unction i and only i the unction is one-to-one. A one-to-one unction is one whose graph passes oth the vertical line test and the horizontal line test. Such a unction is said to e injective. A unction is injective i it maps distinct x in the domain to distinct y in the codomain, such that (x) = y. Put another way, is injective i (a) = () implies a = (or a implies (a) ()), or any a, in the domain. 1 y y = x x The inverse o a relation given y a deining equation is ound y interchanging the x and y variales and solving or y. The latter part o this process can oten e challenging. I the intent is to have only an inverse unction result, the domain must oten e restricted in some ashion in order that the original unction is one-to-one (as descried aove). Logarithms The logarithm o a positive numer N to a given ase (written log N) is the exponent o the power to which must e raised to produce N. It is understood that must e positive and not equal to 1. I ase is e (the ase or natural logarithms) then it is written lnn which is understood to e log e N. I the ase is 10, it is called a common logarithm and logn is understood to e log 10 N. Fundamental Laws or Logarithms log (PQR) = log P + log Q + log R log ( P Q ) = log P + log Q (provide that Q is not zero) log (P n ) = nlog P log ( n P ) = 1 n log P log P = log Q only i P = Q log ( n ) = n log () = 1 log (1) = 0 log N ln N log x = x log N = = (called the change o ase ormula) log ln log N = log 1 1 N log N = log N and we change rom exponential to logarithmic orm using: log x = y x = y
Exponential Functions The exponential unction (one o the most important unctions in mathematics) is written as y = e x, where e equals approximately.7188183 and is the ase o the natural logarithm. The exponential unction is nearly lat (climing slowly) or negative values o x, clims quickly or positive values o x, and equals 1 when x is equal to 0. The eature which makes this unction so useul in mathematics and a wide variety o applications in science technology, economics, geology, etc., is the act that its y value always equals the slope at that point. Sometimes, the term exponential unction is used or unctions o the orm y = ka x, where a, called the ase, is any positive real numer. As a unction o the real variale x, the graph o y = e x is always positive (aove the x axis) and increasing (viewed let-to-right). It never touches the x axis, although it gets aritrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse unction, th e natural logarithm, y = ln(x), is deined or all positive x. The ase o this unction, e, is called Euler s numer, and can e evaluated in a variety o ways. The most common are: 1 1 1 1 1 e = = + + + +... or n! 0! 1!! 3! The exponential laws correspond to most o the laws o logarithms given aove, namely: n= 0 1 e = lim 1+ x n. a 0 = 1, a 0 a 1 = a a x a y = a x+y (a x ) y = a xy a 1 = 1 a Trigonometric Functions 1 1 x x = a x = a a n n n a x x = (a) x a ( a) n = = a Inverse trigonometric unctions (such as sin 1 x or tan 1 x) are unctions that return the angle orm its given trigonometric ratio. The inverse trigonometric unctions are multivalued. For example, there are multiple values o x such that y = sinx, so sin 1 y is not uniquely deined unless a principal value is deined. Such principal values are sometimes denoted with a capital letter so, or example, the principal value o the inverse sine, sin 1 x, may e variously denoted Sin 1 x or y arcsinx. On the other hand, the notation sin 1 x is also commonly used denote either the principal value or any quantity whose sine is x. Worse still, the principal value and multiple valued notations are sometimes reversed, with sin 1 x denoting the principal value and arcsinx denoting the multivalued unctions. Dierent conventions are possile or the domain and range o these unctions or the purpose o keeping them as single-valued unctions; the most in use are illustrated elow. Function name Function Domain Range inverse sine [ 1, 1] 1 1 π, π inverse cosine [ 1, 1] [0, π] inverse tangent (, ) 1 1 π, π inverse cosecant (, ) inverse secant (, ) inverse cotangent (, ) 1 1 π,0 or 0, π 1 1 0, π or ππ, 1 1 - π,0 or 0, π
Napier's ones are an aacus invented y John Napier (orn in Merchiston Tower, in 1550) Edinurgh, or calculation o products and quotients o numers. Also called Radology (rom Greek ραβδoς [rados], rod and λóγoς [logos], word). Napier pulished his invention o the rods in a work printed in Edinurgh, Scotland, at the end o 1617 entitled Radologiæ. Using the multiplication tales emedded in the rods, multiplication can e reduced to addition operations and division to sutractions. More advanced use o the rods can even extract square roots. Note that Napier's ones are not the same as logarithms, with which Napier's name is also associated. The aacus consists o a oard with a rim; the user places Napier's rods in the rim to conduct multiplication or division. The oard's let edge is divided into 9 squares, holding the numers 1 to 9. The Napier's rods consist o strips o wood, metal or heavy cardoard. Napier's ones are three dim ensional, square in cr oss section, with our dierent rods engraved on each one. A set o such ones might e enclosed in a convenient carrying case. A rod's surace comprises 9 squares, and each square, except or the top one, comprises two halves divided y a diagonal line. The irst square o each rod holds a single-digit, and the other squares hold this numer's doule, triple, quadruple and so on until the last square contains nine times the numer in the top square. The digits o each product are written one to each side o the diagonal; numers less than 10 occupy the lower triangle, with a zero in the top hal. A set consists o 9 rods corresponding to digits 1 to 9. The igure additionally shows the rod 0; although or ovious reasons it is not necessary or calculations.
Slide Rules In 1614, John Napier discovered the logarithm which made it possile to perorm multiplications and divisions y addition and sutraction. (ie: a* = 10^(log(a)+log()) and a/ = 10^(log(a)-log()).) This was a great time saver ut there was still quite a lot o work required. The mathematician had to look up two logs, add them together and then look or the numer whose log was the sum. Edmund Gunter soon reduced the eort y drawing a numer line in which the positions o numers were proportional to their logs. The scale started at one ecause the log o one is zero. Two numers could e multiplied y measuring the distance rom the eginning o the scale to one actor with a pair o dividers, then moving them to start at the other actor and reading the numer at the comined distance. The yellow spots are rass inserts to provide wear resistance at commonly used points. Soon aterwards, William Oughtred simpliied things urther y taking two Gunter's lines and sliding them relative to each other thus eliminating the dividers. In the years that ollowed, other people reined Oughtred's design into a sliding ar held in place etween two other ars. Circular slide rules and cylindrical/spiral slide rules also appeared quickly. The cursor appeared on the earliest circular models ut appeared much later on straight versions. By the late 17th century, the slide rule was a common instrument with many variations. It remained the tool o choice or many or the next three hundred years. While great aids, slide rules were not particularly intuitive or eginners. A 1960 Pickett manual said: "When people have diiculty in learning to use a slide rule, usually it is not ecause the instrument is diicult to use. The reason is likely to e that they don't understand the mathematics on which the instrument is ased, or the ormulas they are trying to evaluate.
Some slide rule manuals contain relatively exhaustive explanations o the theory underlying the operations. In this manual it is assumed that the theory o exponents, o logarithms, o trigonometry, and o the slide rule is known to the reader, or will e recalled or studied y reerence to ormal textooks on these sujects." A 1948 Stanley manual expressed a somewhat dierent opinion: "The principles o logarithmic calculators are too well known to those likely to e interested or it to e necessary to enlarge upon the suject here, especially as it is asolutely unnecessary to have any knowledge o the suject to use the calculator"... "Anyone can calculate with the Fuller ater a rie study o the ollowing instructions without any mathematical knowledge whatever." Another interesting quote rom the same Pickett manual: "A computer who must make many diicult calculations usually has a slide rule close at hand." In 1960, "computer" was still understood to e a person who computed. By contrast, a recent dictionary egins the only deinition o "computer" with "An electronic machine..." Some Slide Rule Terms Mannheim A standard single-ace rule with scales to solve prolems in multiplication, division, squares, square roots, reciprocals, trigonometry and logarithms. Polyphase Like a Mannheim ut added a scale or cues and cue roots and an inverted C scale (CI) to make certain prolems easier to solve. (Some manuacturers used Mannheim and Polyphase interchangealy.) Phillips The single sided rule similar to a Polyphase ut with and inverted A scale (typically laeled R) instead o an inverted C scale (CI). Duplex A doule-aced rule. Typically added three olded scales (CF, CIF, DF) to those o the Polyphase rule to make many prolems easier to solve. Trig A rule with scales or solving trigonometry prolems (S, ST, T). Decitrig A rule with the trigonometric scales (S, ST, T) marked in degrees and tenths o a degree. Dual Base A rule with that read oth common and natural logs. Log Log A rule with scales or raising numers to powers. (Scales usually started with LL) Vector A rule with hyperolic unctions. Cominations The aove terms were oten comined on more complex rules like Polyphase Duplex Decitrig. (In this case the doule-sided duplex overrode the single sided assumption o Polyphase.)
Families o Powers Exponent 0 1 3 4 5 6 7 8 9 10 11 1 1 1 1 1 1 1 1 1 1 1 1 1 4 8 16 3 64 18 56 51 104 048 1 3 9 7 81 43 79 187 B 1 4 16 64 56 104 4096 A 1 5 5 15 65 315 S 1 6 36 16 196 7776 E 1 7 49 343 401 16807 1 8 64 51 4096 3768 1 9 81 79 6561 59049 1 10 100 1000 10000 100000 I y = sin 1 x ind cosy and tany. I y = sin 1 5 3 ind cosy and tany. Find the inverse o y = 4x 8. Restrict the domain o y = 4x 8 so that its inverse is a unction. Sketch the graph o the inverse o the unction y = (x) illustrated in the diagram shown elow. y y = x x