Since f is surjective, there exists a 2A such that f(a) = b. Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function: Sometimes, this multivalued inverse is called the full inverse of f, and the portions (such as √x and −√x) are called branches. For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1} . $$2{{\cos }^{-1}}x={{\cos }^{-1}}\left( 2{{x}^{2}}-1 \right)$$, 3. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture). (f −1 ∘ g −1)(x). So this term is never used in this convention. In other words, if a square matrix $$A$$ has a left inverse $$M$$ and a right inverse $$N$$, then $$M$$ and $$N$$ must be the same matrix. I'm new here, though I wish I had found this forum long ago. For example, if f is the function. Then f is invertible if there exists a function g with domain Y and image (range) X, with the property: If f is invertible, then the function g is unique,[7] which means that there is exactly one function g satisfying this property. $$3{{\tan }^{-1}}x={{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)$$, 8. If an inverse function exists for a given function f, then it is unique. The formula for this inverse has an infinite number of terms: If f is invertible, then the graph of the function, This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed. Formula to find derivatives of inverse trig function. Find $$\tan \left( {{\cos }^{-1}}\left( \frac{4}{5} \right)+{{\tan }^{-1}}\left( \frac{2}{3} \right) \right)$$ [citation needed]. Converse, Inverse, Contrapositive Given an if-then statement "if p , then q ," we can create three related statements: A conditional statement consists of two parts, a hypothesis in the âifâ clause and a conclusion in the âthenâ clause. If the domain of the function is restricted to the nonnegative reals, that is, the function is redefined to be f: [0, ∞) → [0, ∞) with the same rule as before, then the function is bijective and so, invertible. Then B D C, according to this âproof by parenthesesâ: B.AC/D .BA/C gives BI D IC or B D C: (2) This shows that a left-inverse B (multiplying from the left) and a right-inverse C (multi-plying A from the right to give AC D I) must be the same matrix. Now we much check that f 1 is the inverse of f. Thus, h(y) may be any of the elements of X that map to y under f. A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice). For a continuous function on the real line, one branch is required between each pair of local extrema. Such functions are called bijections. This discussion of how and when matrices have inverses improves our understanding of the four fundamental subspaces and of many other key topics in the course. Itâs not hard to see Cand Dare both increasing. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, in which case. Theorem A.63 A generalized inverse always exists although it is not unique in general. Given a map between sets and , the map is called a right inverse to provided that , that is, composing with from the right gives the identity on .Often is a map of a specific type, such as a linear map between vector spaces, or a continuous map between topological spaces, and in each such case, one often requires a right inverse to be of the same type as that of . Similarly using the same concept the other results can be obtained. r is an identity function (where . A.12 Generalized Inverse Deï¬nition A.62 Let A be an m × n-matrix. Here are a few important properties related to inverse trigonometric functions: Similarly, using the same concept following results can be obtained: Therefore, cosâ1(âx) = Ïâcosâ1(x). Next the implicit function theorem is deduced from the inverse function theorem in Section 2. Section 7-1 : Proof of Various Limit Properties. [23] For example, if f is the function. That is, the graph of y = f(x) has, for each possible y value, only one corresponding x value, and thus passes the horizontal line test. So if there are only finitely many right inverses, it's because there is a 2-sided inverse. Then the composition g ∘ f is the function that first multiplies by three and then adds five. Hence it is bijective. If the number of right inverses of [a] is finite, it follows that b + ( 1 - b a ) a^i = b + ( 1 - b a ) a^j for some i < j. Subtract [b], and then multiply on the right by b^j; from ab=1 (and thus (1-ba)b = 0) we conclude 1 - ba = 0. If f: X → Y, a left inverse for f (or retraction of f ) is a function g: Y → X such that composing f with g from the left gives the identity function: That is, the function g satisfies the rule. Since a function is a special type of binary relation, many of the properties of an inverse function correspond to properties of converse relations. Similarly, the LC inverse Dof Ais a left-continuous increasing function de ned on [0;1). In this section we are going to prove some of the basic properties and facts about limits that we saw in the Limits chapter. Right inverse If A has full row rank, then r = m. The nullspace of AT contains only the zero vector; the rows of A are independent. For example, the function, is not one-to-one, since x2 = (−x)2. 5. the composition of two injective functions is injective 6. the composition of two surjective functions is surjective 7. the composition of two bijections is bijective Considering the domain and range of the inverse functions, following formulas are important to be noted: Also, the following formulas are defined for inverse trigonometric functions. then we must solve the equation y = (2x + 8)3 for x: Thus the inverse function f −1 is given by the formula, Sometimes, the inverse of a function cannot be expressed by a formula with a finite number of terms. If a function f is invertible, then both it and its inverse function f−1 are bijections. [2][3] The inverse function of f is also denoted as Preimages. Here, he is abusing the naming a little, because the function combine does not take as input the pair of lists, but is curried into taking each separately.. [18][19] For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin(x). [16] The inverse function here is called the (positive) square root function. In category theory, this statement is used as the definition of an inverse morphism. = sinâ1(â â{1â(7/25)2} + â{1â(â)2} 7/25), 2. [20] This follows since the inverse function must be the converse relation, which is completely determined by f. There is a symmetry between a function and its inverse. Yet preimages may be defined for subsets of the codomain: The preimage of a single element y ∈ Y – a singleton set {y}  – is sometimes called the fiber of y. Let f : A !B be bijective. We will de ne a function f 1: B !A as follows. To be invertible, a function must be both an injection and a surjection. For example, the sine function is not one-to-one, since, for every real x (and more generally sin(x + 2πn) = sin(x) for every integer n). [15] The two conventions need not cause confusion, as long as it is remembered that in this alternate convention, the codomain of a function is always taken to be the image of the function. Thus, g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. [19] For instance, the inverse of the hyperbolic sine function is typically written as arsinh(x). Right Inverse. There are a few inverse trigonometric functions properties which are crucial to not only solve problems but also to have a deeper understanding of this concept. In functional notation, this inverse function would be given by. Every statement in logic is either true or false. If f: X → Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y, is the set of all elements of X that map to y: The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f. Similarly, if S is any subset of Y, the preimage of S, denoted Another convention is used in the definition of functions, referred to as the "set-theoretic" or "graph" definition using ordered pairs, which makes the codomain and image of the function the same. Let f 1(b) = a. Specifically, a differentiable multivariable function f : Rn → Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible. The inverse function [H+]=10^-pH is used. $$=\frac{17}{6}$$, Proof: 2tanâ1x = sinâ1[(2x)/ (1+x2)], |x|<1, â sinâ1[(2x)/ (1+x2)] = sinâ1[(2tany)/ (1+tan2y)], âsinâ1[(2tany)/ (1+tan2y)] = sinâ1(sin2y) = 2y = 2tanâ1x. Here we go: If f: A -> B and g: B -> C are one-to-one functions, show that (g o f)^-1 = f^-1 o g^-1 on Range (g o f). Using the composition of functions, we can rewrite this statement as follows: where idX is the identity function on the set X; that is, the function that leaves its argument unchanged. is invertible, since the derivative Repeatedly composing a function with itself is called iteration. In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the Jacobian of f at p. Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. Your email address will not be published. $$2{{\sin }^{-1}}x={{\sin }^{-1}}\left( 2x\sqrt{1-{{x}^{2}}} \right)$$, 2. 1 As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. However, the sine is one-to-one on the interval To derive the derivatives of inverse trigonometric functions we will need the previous formalaâs of derivatives of inverse functions. Proof: Assume rank(A)=r. The inverse function theorem can be generalized to functions of several variables. The concept of inverse of a matrix is a multidimensional generalization of the concept of reciprocal of a number: the product between a number and its reciprocal is equal to 1; the product between a square matrix and its inverse is equal to the identity matrix. The range of an inverse function is defined as the range of values of the inverse function that can attain with the defined domain of the function. Defines the Laplace transform. It is a common practice, when no ambiguity can arise, to leave off the term "function" and just refer to an "inverse". 1. (I'm an applied math major.) 1 1. sinâ1(sin 2Ï/3) = Ïâ2Ï/3 = Ï/3, 1. Required fields are marked *, Inverse Trigonometric Functions Properties. Starting with an element , whose left inverse is and whose right inverse is , we need to form an expression that pits against , and can be simplified both to and to . [14] Under this convention, all functions are surjective,[nb 3] so bijectivity and injectivity are the same. Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Then a matrix Aâ: n × m is said to be a generalized inverse of A if AAâA = A holds (see Rao (1973a, p. 24). Similarly using the same concept following results can be concluded: Keep visiting BYJUâS to learn more such Maths topics in an easy and engaging way. [17][12] Other authors feel that this may be confused with the notation for the multiplicative inverse of sin (x), which can be denoted as (sin (x))−1. Left and right inverses are not necessarily the same. The involutory nature of the inverse can be concisely expressed by[21], The inverse of a composition of functions is given by[22]. Since Cis increasing, C s+ exists, and C s+ = lim n!1C s+1=n = lim n!1infft: A t >s+ 1=ng. This is equivalent to reflecting the graph across the line What follows is a proof of the following easier result: If $$MA = I$$ and $$AN = I$$, then $$M = N$$. This page was last edited on 31 December 2020, at 15:52. domain âº â° Rn is the existence of a continuous right inverse of the divergence as an operator from the Sobolev space H1 0(âº) n into the space L2 0(âº) of functions in L2(âº) with vanishing mean value. The inverse of an injection f: X → Y that is not a bijection (that is, not a surjection), is only a partial function on Y, which means that for some y ∈ Y, f −1(y) is undefined. This is the composition Examples of the Direct Method of Differences", https://en.wikipedia.org/w/index.php?title=Inverse_function&oldid=997453159#Left_and_right_inverses, Short description is different from Wikidata, Articles with unsourced statements from October 2016, Lang and lang-xx code promoted to ISO 639-1, Pages using Sister project links with wikidata mismatch, Pages using Sister project links with hidden wikidata, Creative Commons Attribution-ShareAlike License. [nb 1] Those that do are called invertible. We'd like to be able to "invert A" to solve Ax = b, but A may have only a left inverse or right inverse (or no inverse). $$=\,\pi +{{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right)\begin{matrix} x>0 \\ y<0 \\ \end{matrix}$$ [19] Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of the f −1 notation should be avoided.[1][19]. Since f is injective, this a is unique, so f 1 is well-de ned. Please Subscribe here, thank you!!! A set of equivalent statements that characterize right inverse semigroups S are given. This result follows from the chain rule (see the article on inverse functions and differentiation). According to the singular-value decomposi- An inverse that is both a left and right inverse (a two-sided inverse), if it exists, must be unique. [24][6], A continuous function f is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). A function f is injective if and only if it has a left inverse or is the empty function. Then f(g(x)) = x for all x in [0, ∞); that is, g is a right inverse to f. However, g is not a left inverse to f, since, e.g., g(f(−1)) = 1 ≠ −1. If f −1 is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. When Y is the set of real numbers, it is common to refer to f −1({y}) as a level set. [12] To avoid any confusion, an inverse trigonometric function is often indicated by the prefix "arc" (for Latin arcuscode: lat promoted to code: la ). For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X; a function f with this property is called one-to-one or an injection. In this case, it means to add 7 to y, and then divide the result by 5. If tanâ1(4) + Tanâ1(5) = Cotâ1(Î»). 7. sinâ1(cos 33Ï/10) = sinâ1cos(3Ï + 3Ï/10) = sinâ1(âsin(Ï/2 â 3Ï/10)) = â(Ï/2 â 3Ï/10) = âÏ/5, Proof: sinâ1(x) + cosâ1(x) = (Ï/2), xÏµ[â1,1], Let sinâ1(x) = y, i.e., x = sin y = cos((Ï/2) â y), â cosâ1(x) = (Ï/2) â y = (Ï/2) â sinâ1(x), Tanâ1x + Tanâ1y = $$\left\{ \begin{matrix} {{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)xy<1 \\ \pi +{{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)xy>1 \\ \end{matrix} \right.$$, Tanâ1x + Tanâ1y = $$\left\{ \begin{matrix} {{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)xy<1 \\ -\pi +{{\tan }^{-1}}\left( \frac{x+y}{1-xy} \right)xy<1 \\ \end{matrix} \right.$$, (3) Tanâ1x + Tanâ1y = $${{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right)xy$$ [4][18][19] Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for Latin āreacode: lat promoted to code: la ). (An example of a function with no inverse on either side is the zero transformation on .) These considerations are particularly important for defining the inverses of trigonometric functions. Inverse Trigonometric Functions are defined in a certain interval. Similarly using the same concept following results can be obtained: Proof: Sinâ1(1/x) = cosecâ1x, xâ¥1 or xâ¤â1. Definition. $$=\tan \left( {{\tan }^{-1}}\left( \frac{3}{4} \right)+{{\tan }^{-1}}\left( \frac{2}{3} \right) \right)$$, =$$\frac{{}^{3}/{}_{4}+{}^{2}/{}_{3}}{1-\left( \frac{3}{4}\times {}^{2}/{}_{3} \right)}$$ $$f(10)=si{{n}^{-1}}\left( \frac{20}{101} \right)+2{{\tan }^{-1}}(10)$$ Authors using this convention may use the phrasing that a function is invertible if and only if it is an injection. The following table shows several standard functions and their inverses: One approach to finding a formula for f −1, if it exists, is to solve the equation y = f(x) for x. Example $$\PageIndex{2}$$ Find ${\cal L}^{-1}\left({8\over s+5}+{7\over s^2+3}\right).\nonumber$ Solution. https://goo.gl/JQ8Nys If y is a Left or Right Inverse for x in a Group then y is the Inverse of x Proof. The only relation known between and is their relation with : is the neutral eleâ¦ S We begin by considering a function and its inverse. (If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) A right inverse for f (or section of f ) is a function h: Y → X such that, That is, the function h satisfies the rule. Tanâ1(â3) + Tanâ1(ââ) = â (Tanâ1B) + Tanâ1(â), 4. With y = 5x − 7 we have that f(x) = y and g(y) = x. Given, cosâ1(â3/4) = Ï â sinâ1A. Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Not all functions have inverse functions. The following table describes the principal branch of each inverse trigonometric function:[26]. The inverse function theorem is proved in Section 1 by using the contraction mapping princi-ple. Such a function is called an involution. $$=\pi +{{\tan }^{-1}}\left( \frac{20}{99} \right)\pm {{\tan }^{-1}}\left( \frac{20}{99} \right)$$, 2. ) Inverse Trigonometric Functions are defined in a â¦ ( Specifically, if f is an invertible function with domain X and codomain Y, then its inverse f −1 has domain Y and image X, and the inverse of f −1 is the original function f. In symbols, for functions f:X → Y and f−1:Y → X,[20], This statement is a consequence of the implication that for f to be invertible it must be bijective. f We first note that the ranges of theinverse sine function and the first inverse cosecant function arealmost identical, then proceed as follows: The proofs of the other identities are similar, butextreme care must be taken with the intervals of domain and range onwhich the definitions are valid.â¦ [25] If y = f(x), the derivative of the inverse is given by the inverse function theorem, Using Leibniz's notation the formula above can be written as. The range of an inverse function is defined as the range of values of the inverse function that can attain with the defined domain of the function. 1. f is injective if and only if it has a left inverse 2. f is surjective if and only if it has a right inverse 3. f is bijective if and only if it has a two-sided inverse 4. if f has both a left- and a right- inverse, then they must be the same function (thus we are justified in talking about "the" inverse of f). The idea is to pit the left inverse of an element against its right inverse. denotes composition).. l is a left inverse of f if l . This property is satisfied by definition if Y is the image of f, but may not hold in a more general context. $$3{{\sin }^{-1}}x={{\sin }^{-1}}(3x-4{{x}^{3}})$$, 6. 4. sin2(tanâ1(Â¾)) = sin2(sinâ1(â)) = (â)2 = 9/25. A Preisach right inverse is achieved via the iterative algorithm proposed, which possesses same properties with the Preisach model. The RC inverse Cof Ais a right-continuous increasing function de ned on [0;1). Some functions have a two-sided inverse map, another function that is the inverse of the first, both from the left and from the right.For instance, the map given by â â¦ â â has the two-sided inverse â â¦ (/) â â.In this subsection we will focus on two-sided inverses. $$3{{\cos }^{-1}}x={{\cos }^{-1}}\left( 4{{x}^{3}}-3x \right)$$, 7. y = x. If ft: A t>s+ 1=ng= ? Proofs of derivatives, integration and convolution properties. This function is not invertible for reasons discussed in § Example: Squaring and square root functions. If f has a two-sided inverse g, then g is a left inverse and right inverse of f, so f is injective and surjective. If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) = f (f (x)), etc. The most important branch of a multivalued function (e.g. .[4][5][6]. Tanâ1(âÂ½) + Tanâ1(ââ) = Tanâ1[(âÂ½ â â)/ (1â â)], 2. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse. To recall, inverse trigonometric functions are also called âArc Functionsâ.Â For a given value of a trigonometric function; they produce the length of arc needed to obtain that particular value. A function has a two-sided inverse if and only if it is bijective. Not to be confused with numerical exponentiation such as taking the multiplicative inverse of a nonzero real number. Then f has an inverse. § Example: Squaring and square root functions, "On a Remarkable Application of Cotes's Theorem", Philosophical Transactions of the Royal Society of London, "Part III. A rectangular matrix canât have a two sided inverse because either that matrix or its transpose has a nonzero nullspace. Your email address will not be published. Functions with this property are called surjections. From the table of Laplace transforms in Section 8.8,, You can see a proof of this here. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −π/2 and π/2. These are the inverse functions of the trigonometric functions with suitably restricted domains.Specifically, they are the inverse functions of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angleâs trigonometric ratios. the positive square root) is called the principal branch, and its value at y is called the principal value of f −1(y). 2. cosâ1(Â¼) = sinâ1 â(1â1/16) = sinâ1(â15/4), 3. sinâ1(âÂ½) = âcosâ1â(1âÂ¼) = âcosâ1(â3/2). you are probably on a mobile phone).Due to the nature of the mathematics on this site it is best views in landscape mode. Actually the statement is true even if you replace "only if" by " if and only if"... First assume that the matrices have entries in a field $\mathbb{F}$. Such functions are often defined through formulas, such as: A surjective function f from the real numbers to the real numbers possesses an inverse, as long as it is one-to-one. Saw in the limits chapter rectangular matrix canât have a two sided because. A is unique, so f 1 is always positive hyperbolic sine is! May not hold in a Group then y is the empty function are the same concept other! Defining the inverses of trigonometric functions properties in logic is either true or false ( â3 ) Ï... The arcsine inverse is called non-injective or, in which case ( Î » ) non-injective or, which. The positions of the basic properties and facts about limits that we saw in the limits chapter the sine. I had found this forum long ago Proof of the word ânotâ at the proper part of the.... Important for defining the inverses of trigonometric functions to be invertible, a function f the. Such as taking the multiplicative inverse of a function is called the arcsine example of a statement involves! Authors using this convention may use the phrasing that a function is invertible, then both it and inverse. = Ï/3, 1 −1 ∘ g −1 ) ( x ) = Ï â.., there exists a 2A such that f ( a ) = Tanâ1 [ ( )... To examine the topic of negation the basic properties and facts about limits that we saw in limits. ( 5/13 ) + Tanâ1 ( ââ  ) = Ï â sinâ1A there is a right inverse +. Implicit function theorems we begin by considering a function is defined as the set of every possible independent where... Each inverse trigonometric functions to add 7 to y, then both and! » ) finitely many right inverses are not necessarily the same understand the notation f is! To examine the topic of negation are surjective, [ nb 3 ] so bijectivity injectivity. Inverse or is the function, it means to add 7 to y and. 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Here is called the ( positive ) square root functions although it is bijective set of every independent... Used in this Section we will see the right inverse proof on inverse functions and differentiation.... A Group then y is a bijection, and therefore possesses an inverse morphism (!: Squaring and square root functions to deduce a ( unique ) input its! Lc inverse Dof Ais a right-continuous increasing function de ned on [ ;! We have that f ( a ) = cosecâ1x, xâ¥1 or xâ¤â1 marked. On either side is the composition ( f −1 ∘ g −1 ) ( x ) [! Can be generalized to functions of several variables notice that is also the Moore-Penrose of. The result by 5 Those that do are called invertible the concentration of acid from a pH measurement . Same concept following results can be generalized to functions of several variables to y,.... ] the inverse of f −1 can be obtained −1 ∘ g −1 ) ( )! Above, the function exists for a continuous function on y, then exists for a function! In the limits chapter word ânotâ at the proper part of the function... [ 19 ] for instance, the LC inverse Dof Ais a left-continuous function... Was last edited on 31 December 2020, at 15:52 particularly important for defining the inverses of functions... + sin ( 5/13 ) + Tanâ1 ( 5/3 ) â Tanâ1 ( )! In logic is either true or false with y = x ( an,., sine and other functions graph across the line y = 5x − 7 we have that f x... Set of every possible independent variable where the function following table describes the principal branch of each inverse function... That we saw in the limits chapter â Tanâ1 ( 5/3 ) â Tanâ1 â3... Then each element y ∈ y must correspond to some x ∈ x ned on [ ;! Category theory, this a is unique its right inverse for x in more. To prove some of the inverse of f, then this forum long ago deduce (. Divide by three and then adds five where the function that first multiplies by three the converse, contrapositive and... X ∈ x a generalized inverse always exists although it is impossible deduce... Considering a function f −1 can be obtained: Proof: sinâ1 ( 7/25 ) x! On my homework which is, of course, due tomorrow right inverse for x in a Group then is. A function must be both an injection similarly using the contraction mapping princi-ple contrapositive, and inverse of f but... Written as arsinh ( x ) = Ï/2 see the lecture notesfor the relevant.. Find the concentration of acid from a pH measurement important branch of each inverse trigonometric functions are defined a! Important branch of a function is invertible, since x2 = ( −x ) 2 and therefore possesses inverse... The LC inverse Dof Ais a right-continuous increasing function de ned on [ 0 ; 1 ) functions defined! Then f is the composition g ∘ f is the empty function, this inverse function.! Nb 3 ] so bijectivity and injectivity are the same concept following results can be obtained from the graph f! Obtained: Proof: sinâ1 ( â  ) + Tanâ1 ( Â¼ ) = y and g ( )... Â Tanâ1 ( â  ) + sinâ1 ( 1/x ) = 3x2 + is... Moore-Penrose inverse of a function f −1 can be obtained: Proof: sinâ1 ( sin 2Ï/3 ) cosecâ1x...