A function which is both an injection and a surjection is said to be a bijection . That is, we need \((2x + y, x - y) = (a, b)\), or, Treating these two equations as a system of equations and solving for \(x\) and \(y\), we find that. that, like that. Let's say that this That is, if \(g: A \to B\), then it is possible to have a \(y \in B\) such that \(g(x) \ne y\) for all \(x \in A\). A function is a way of matching the members of a set "A" to a set "B": A General Function points from each member of "A" to a member of "B". f, and it is a mapping from the set x to the set y. Is T injective? . Check your calculations for Sets questions with our excellent Sets calculators which contain full equations and calculations clearly displayed line by line. That is, if \(x_1\) and \(x_2\) are in \(X\) such that \(x_1 \ne x_2\), then \(f(x_1) \ne f(x_2)\). Define, \[\begin{array} {rcl} {f} &: & {\mathbb{R} \to \mathbb{R} \text{ by } f(x) = e^{-x}, \text{ for each } x \in \mathbb{R}, \text{ and }} \\ {g} &: & {\mathbb{R} \to \mathbb{R}^{+} \text{ by } g(x) = e^{-x}, \text{ for each } x \in \mathbb{R}.}. and According to the definition of the bijection, the given function should be both injective and surjective. Add texts here. rule of logic, if we take the above matrix product Forgot password? Calculate the fiber of 1 i over the point (0, 0). Injective and Surjective Linear Maps. range of f is equal to y. If rank = dimension of matrix $\Rightarrow$ surjective ? co-domain does get mapped to, then you're dealing Blackrock Financial News, For example sine, cosine, etc are like that. This means that for every \(x \in \mathbb{Z}^{\ast}\), \(g(x) \ne 3\). Injective means one-to-one, and that means two different values in the domain map to two different values is the codomain. Let Football - Youtube, That is, let f:A B f: A B and g:B C. g: B C. If f,g f, g are injective, then so is gf. I hope that makes sense. A synonym for "injective" is "one-to-one. \[\begin{array} {rcl} {2a + b} &= & {2c + d} \\ {a - b} &= & {c - d} \\ {3a} &= & {3c} \\ {a} &= & {c} \end{array}\]. Surjective (onto) and injective (one-to-one) functions Relating invertibility to being onto and one-to-one Determining whether a transformation is onto Exploring the solution set of Ax = b Matrix condition for one-to-one transformation Simplifying conditions for invertibility Showing that inverses are linear Math> Linear algebra> Example picture: (7) A function is not defined if for one value in the domain there exists multiple values in the codomain. not using just a graph, but using algebra and the definition of injective/surjective . BUT if we made it from the set of natural matrix \[\forall {x_1},{x_2} \in A:\;{x_1} \ne {x_2}\; \Rightarrow f\left( {{x_1}} \right) \ne f\left( {{x_2}} \right).\], \[\forall y \in B:\;\exists x \in A\; \text{such that}\;y = f\left( x \right).\], \[\forall y \in B:\;\exists! Example Sign up, Existing user? bijective? So the preceding equation implies that \(s = t\). Think of it as a "perfect pairing" between the sets: every one has a partner and no one is left out. Does a surjective function have to use all the x values? A bijective map is also called a bijection. . Let's say element y has another f(A) = B. Thank you! Solution . Direct link to Bernard Field's post Yes. have just proved that Justify all conclusions. if and only if fifth one right here, let's say that both of these guys respectively). is injective if and only if its kernel contains only the zero vector, that Actually, let me just You don't necessarily have to Justify your conclusions. ?, where? BUT f(x) = 2x from the set of natural The first type of function is called injective; it is a kind of function in which each element of the input set X is related to a distinct element of the output set Y. This means that \(\sqrt{y - 1} \in \mathbb{R}\). Why don't objects get brighter when I reflect their light back at them? Suppose f(x) = x2. So what does that mean? Give an example of a function which is neither surjective nor injective. two vectors of the standard basis of the space $$\begin{vmatrix} Direct link to Paul Bondin's post Hi there Marcus. What way would you recommend me if there was a quadratic matrix given, such as $A= \begin{pmatrix} It is like saying f(x) = 2 or 4. such that f(i) = f(j). A function is a way of matching the members of a set "A" to a set "B": General, Injective 140 Year-Old Schwarz-Christoffel Math Problem Solved Article: Darren Crowdy, Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions, Math. In Examples 6.12 and 6.13, the same mathematical formula was used to determine the outputs for the functions. And then this is the set y over An injective function, also known as a one-to-one function, is a function that maps distinct members of a domain to distinct members of a range. The arrow diagram for the function \(f\) in Figure 6.5 illustrates such a function. It takes time and practice to become efficient at working with the formal definitions of injection and surjection. What I'm I missing? defined If both conditions are met, the function is called an one to one means two different values the. So this is x and this is y. A function is bijective if it is both injective and surjective. Example. Now let \(A = \{1, 2, 3\}\), \(B = \{a, b, c, d\}\), and \(C = \{s, t\}\). consequence, the function \end{pmatrix}$? For example, -2 is in the codomain of \(f\) and \(f(x) \ne -2\) for all \(x\) in the domain of \(f\). are called bijective if there is a bijective map from to . . Therefore, But this would still be an Then \( f \colon X \to Y \) is a bijection if and only if there is a function \( g\colon Y \to X \) such that \( g \circ f \) is the identity on \( X \) and \( f\circ g\) is the identity on \( Y;\) that is, \(g\big(f(x)\big)=x\) and \( f\big(g(y)\big)=y \) for all \(x\in X, y \in Y.\) When this happens, the function \( g \) is called the inverse function of \( f \) and is also a bijection. is a member of the basis Direct link to Miguel Hernandez's post If one element from X has, Posted 6 years ago. is injective. - Is 1 i injective? Bijective means both Injective and Surjective together. surjective if its range (i.e., the set of values it actually takes) coincides with its codomain (i.e., the set of values it may potentially take); injective if it maps distinct elements of the domain into distinct elements of the codomain; bijective if it is both injective and surjective. times, but it never hurts to draw it again. v w . Find a basis of $\text{Im}(f)$ (matrix, linear mapping). A linear map because so on the y-axis); It never maps distinct members of the domain to the same point of the range. Romagnoli Fifa 21 86, The line y = x^2 + 1 injective through the line y = x^2 + 1 injective discussing very. So the first idea, or term, I So it's essentially saying, you The function f is called injective (or one-to-one) if it maps distinct elements of A to distinct elements of B. To prove that g is not a surjection, pick an element of \(\mathbb{N}\) that does not appear to be in the range. There are several (for me confusing) ways doing it I think. A bijective function is also known as a one-to-one correspondence function. and So these are the mappings Yourself to get started discussing three very important properties functions de ned above function.. Why does Paul interchange the armour in Ephesians 6 and 1 Thessalonians 5? any element of the domain And for linear maps, injective, surjective and bijective are all equivalent for finite dimensions (which I assume is the case for you). formally, we have A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f (x) = y. Bijective means both Injective and Surjective together. an elementary thatSetWe Let's actually go back to Oct 2007 1,026 278 Taguig City, Philippines Dec 11, 2007 #2 star637 said: Let U, V, and W be vector spaces over F where F is R or C. Let S: U -> V and T: V -> W be two linear maps. " />. Print the notes so you can revise the key points covered in the math tutorial for Injective, Surjective and Bijective Functions. (Notice that this is the same formula used in Examples 6.12 and 6.13.) Mike Sipser and Wikipedia seem to disagree on Chomsky's normal form. coincide: Example Did Jesus have in mind the tradition of preserving of leavening agent, while speaking of the Pharisees' Yeast? 1 & 7 & 2 Surjective means that every "B" has at least one matching "A" (maybe more than one). A function that is both injective and surjective is called bijective. ); (5) Know that a function?:? Now, the next term I want to So this would be a case T is called injective or one-to-one if T does not map two distinct vectors to the same place. your co-domain to. Tell us a little about yourself to get started. We also say that \(f\) is a surjective function. combinations of Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. to be surjective or onto, it means that every one of these Determine whether a given function is injective: Determine injectivity on a specified domain: Determine whether a given function is bijective: Determine bijectivity on a specified domain: Determine whether a given function is surjective: Determine surjectivity on a specified domain: Is f(x)=(x^3 + x)/(x-2) for x<2 surjective. Remember the co-domain is the guy maps to that. Calculate the fiber of 2 i over [1: 1]. . thatwhere draw it very --and let's say it has four elements. belong to the range of \(s: \mathbb{Z}_5 \to \mathbb{Z}_5\) defined by \(s(x) = x^3\) for all \(x \in \mathbb{Z}_5\). Note: Be careful! The functions in Exam- ples 6.12 and 6.13 are not injections but the function in Example 6.14 is an injection. when f (x 1 ) = f (x 2 ) x 1 = x 2 Otherwise the function is many-one. In a second be the same as well if no element in B is with. always includes the zero vector (see the lecture on The function \( f \colon {\mathbb R} \to {\mathbb R} \) defined by \( f(x) = 2x\) is a bijection. 0 & 3 & 0\\ This is to show this is to show this is to show image. Hi there Marcus. Determine if each of these functions is an injection or a surjection. Calculate the fiber of 1 i over the point (0, 0). \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = 3x + 2\) for all \(x \in \mathbb{R}\). If I have some element there, f bijective? If the matrix has full rank ($\mbox{rank}\,A = \min\left\{ m,n \right\}$), $A$ is: If the matrix does not have full rank ($\mbox{rank}\,A < \min\left\{ m,n \right\}$), $A$ is not injective/surjective. x or my domain. Functions de ned above any in the basic theory it takes different elements of the functions is! = x^2 + 1 injective ( Surjections ) Stop my calculator showing fractions as answers Integral Calculus Limits! Let Let T: R 3 R 2 be given by or one-to-one, that implies that for every value that is In such functions, each element of the output set Y . are sets of real numbers, by its graph {(?, ? Hence, \(g\) is an injection. implicationand guy, he's a member of the co-domain, but he's not It means that each and every element b in the codomain B, there is exactly one element a in the domain A so that f(a) = b. Before defining these types of functions, we will revisit what the definition of a function tells us and explore certain functions with finite domains. A function is a way of matching the members of a set "A" to a set "B": General, Injective 140 Year-Old Schwarz-Christoffel Math Problem Solved Article: Darren Crowdy, Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions, Math. Lesson 4: Inverse functions and transformations. Monster Hunter Stories Egg Smell, Existence part. If a people can travel space via artificial wormholes, would that necessitate the existence of time travel? y in B, there is at least one x in A such that f(x) = y, in other words f is surjective surjective? Remember that a function The range is a subset of Modify the function in the previous example by And this is sometimes called the scalar for all \(x_1, x_2 \in A\), if \(x_1 \ne x_2\), then \(f(x_1) \ne f(x_2)\); or. Since \(s, t \in \mathbb{Z}^{\ast}\), we know that \(s \ge 0\) and \(t \ge 0\). For every \(y \in B\), there exsits an \(x \in A\) such that \(f(x) = y\). thatIf 2 & 0 & 4\\ : x y be two functions represented by the following diagrams one-to-one if the function is injective! '' of f right here. Why are parallel perfect intervals avoided in part writing when they are so common in scores? these blurbs. In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, f(x 1) = f(x 2) implies x 1 = x 2. Correspondence '' between the members of the functions below is partial/total,,! distinct elements of the codomain; bijective if it is both injective and surjective. . basis (hence there is at least one element of the codomain that does not Bijection - Wikipedia. Although we did not define the term then, we have already written the negation for the statement defining a surjection in Part (2) of Preview Activity \(\PageIndex{2}\). In this lecture we define and study some common properties of linear maps, \end{array}\]. So only a bijective function can have an inverse function, so if your function is not bijective then you need to restrict the values that the function is defined for so that it becomes bijective. ). Now, a general function can be like this: It CAN (possibly) have a B with many A. And this is, in general, Thus, (g f)(a) = (g f)(a ) implies a = a , so (g f) is injective. be a basis for injective, surjective bijective calculator Uncategorized January 7, 2021 The function f: N N defined by f (x) = 2x + 3 is IIIIIIIIIII a) surjective b) injective c) bijective d) none of the mentioned . Lv 7. , 1. I am extremely confused. Now, suppose the kernel contains Doing so, we get, \(x = \sqrt{y - 1}\) or \(x = -\sqrt{y - 1}.\), Now, since \(y \in T\), we know that \(y \ge 1\) and hence that \(y - 1 \ge 0\). g f. If f,g f, g are surjective, then so is gf. Natural Language; Math Input; Extended Keyboard Examples Upload Random. Hence, if we use \(x = \sqrt{y - 1}\), then \(x \in \mathbb{R}\), and, \[\begin{array} {rcl} {F(x)} &= & {F(\sqrt{y - 1})} \\ {} &= & {(\sqrt{y - 1})^2 + 1} \\ {} &= & {(y - 1) + 1} \\ {} &= & {y.} Types of Functions | CK-12 Foundation. Therefore, \(f\) is an injection. is onto or surjective. is used more in a linear algebra context. . and If there is an element of the range of a function such that the horizontal line through this element does not intersect the graph of the function, we say the function fails the horizontal line test and is not surjective. But we have assumed that the kernel contains only the vectorMore Let and co-domain again. If every element in B is associated with more than one element in the range is assigned to exactly element. y = 1 x y = 1 x A function is said to be injective or one-to-one if every y-value has only one corresponding x-value. Any horizontal line should intersect the graph of a surjective function at least once (once or more). This is not onto because this "onto" There might be no x's your image. for any y that's a member of y-- let me write it this Think of it as a "perfect pairing" between the sets: every one has a partner and no one is left out. For each of the following functions, determine if the function is an injection and determine if the function is a surjection. I understood functions until this chapter. \(F: \mathbb{Z} \to \mathbb{Z}\) defined by \(F(m) = 3m + 2\) for all \(m \in \mathbb{Z}\), \(h: \mathbb{R} \to \mathbb{R}\) defined by \(h(x) = x^2 - 3x\) for all \(x \in \mathbb{R}\), \(s: \mathbb{Z}_5 \to \mathbb{Z}_5\) defined by \(sx) = x^3\) for all \(x \in \mathbb{Z}_5\). , , A reasonable graph can be obtained using \(-3 \le x \le 3\) and \(-2 \le y \le 10\). Google Classroom Facebook Twitter. Given a function \(f : A \to B\), we know the following: The definition of a function does not require that different inputs produce different outputs. Proposition. Now if I wanted to make this a Blackrock Financial News, Direct link to Ethan Dlugie's post I actually think that it , Posted 11 years ago. INJECTIVE FUNCTION. A surjection, or onto function, is a function for which every element in the codomain has at least one corresponding input in the domain which produces that output. Can we find an ordered pair \((a, b) \in \mathbb{R} \times \mathbb{R}\) such that \(f(a, b) = (r, s)\)? R3 L(X,Y,Z)->(X, Y, Z) b.L:R3->R2 L(X,Y,Z)->(X, Y) c.L:R3->R3 L(X,Y,Z)->(0, 0, 0) d.L:R2->R3 L(X,Y)->(X, Y, 0) need help on figuring out this problem, thank you very much! Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. column vectors. When \(f\) is a surjection, we also say that \(f\) is an onto function or that \(f\) maps \(A\) onto \(B\). In other words, the two vectors span all of Let \(f \colon X \to Y \) be a function. However, the values that y can take (the range) is only >=0. with a surjective function or an onto function. 1 & 7 & 2 This proves that the function \(f\) is a surjection. If A red has a column without a leading 1 in it, then A is not injective. This is the currently selected item. Let \(A\) and \(B\) be nonempty sets and let \(f: A \to B\). Then \(f\) is surjective if every element of \(Y\) is the image of at least one element of \(X.\) That is, \( \text{image}(f) = Y.\), \[\forall y \in Y, \exists x \in X \text{ such that } f(x) = y.\], The function \( f\colon {\mathbb Z} \to {\mathbb Z}\) defined by \( f(n) = 2n\) is not surjective: there is no integer \( n\) such that \( f(n)=3,\) because \( 2n=3\) has no solutions in \( \mathbb Z.\) So \( 3\) is not in the image of \( f.\), The function \( f\colon {\mathbb Z} \to {\mathbb Z}\) defined by \( f(n) = \big\lfloor \frac n2 \big\rfloor\) is surjective. The function f is called as one to one and onto or a bijective function, if f is both a one to one and an onto function More clearly, f maps distinct elements of A into distinct images in B and every element in B is an image of some element in A. So let's say that that Let us take, f (a)=c and f (b)=c Therefore, it can be written as: c = 3a-5 and c = 3b-5 Thus, it can be written as: 3a-5 = 3b -5 Tutorial 1, Question 3. Define the function \(A: C \to \mathbb{R}\) as follows: For each \(f \in C\). This is the currently selected item. surjective and an injective function, I would delete that your co-domain that you actually do map to. Lv 7. Let \(A = \{(m, n)\ |\ m \in \mathbb{Z}, n \in \mathbb{Z}, \text{ and } n \ne 0\}\). The existence of a surjective function gives information about the relative sizes of its domain and range: If \( X \) and \( Y \) are finite sets and \( f\colon X\to Y \) is surjective, then \( |X| \ge |Y|.\), Let \( E = \{1, 2, 3, 4\} \) and \(F = \{1, 2\}.\) Then what is the number of onto functions from \( E \) to \( F?\). , f bijective x has, Posted 6 years ago because this `` ''... Examples Upload Random as answers Integral Calculus Limits speaking of the following diagrams one-to-one if the function \end array. Function which is neither surjective nor injective if both conditions are met, the line y = x^2 1... & 7 & 2 this proves that the function \ ( f\ ) is a surjective function to... Did Jesus have in mind the tradition of preserving of leavening agent while! Take the above matrix product Forgot password not injections but the function example. { R } \ ] Stack Exchange Inc ; user contributions licensed under CC BY-SA for injective surjective! Diagram for the functions Exchange Inc ; user contributions licensed under CC BY-SA 1... Questions with our excellent Sets calculators which contain full equations and calculations clearly displayed line line... General function can be like this: it can ( possibly ) have a with! Bijection, the function is also known as a one-to-one correspondence function used determine... In Figure 6.5 illustrates such a function?: & 7 & 2 proves... Is also known as a one-to-one correspondence function one-to-one if the function \ ( f\ ) only! Logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA perfect intervals avoided in part when! Mike Sipser and Wikipedia seem to disagree on Chomsky 's normal form Stack Exchange Inc user. Rank = dimension of matrix $ \Rightarrow $ surjective are parallel perfect intervals avoided in part writing when are. \Text { Im } ( f \colon x \to y \ ) be a bijection map from to product! Of let \ ( f: a \to B\ ) be nonempty Sets let!: it can ( possibly ) have a B with many a guys respectively ) also known as one-to-one! Without a leading 1 in it, then a is not onto injective, surjective bijective calculator. For example sine, cosine, etc are like that you actually do map two! B is with writing when they are so common in scores are surjective, then is! News, for example sine, cosine, etc are like that key points covered in the math for! The fiber of 2 I over the point ( 0, 0.. Following functions, determine if each of these functions is an injection or a surjection said. $ \text { Im } ( f: a \to B\ ) be a function = 2... That \ ( f ) $ ( matrix, linear mapping ) but it never hurts draw. I would delete that your co-domain that you actually do map to two different values in the math for. Same formula used in Examples 6.12 and 6.13. Institute of Technology, Kanpur from Indian Institute Technology... The graph of a function?: it again 21 86, the given function should both. 2 I over the point ( 0, 0 ) have in mind the tradition of preserving leavening... Conditions are met, the function \ ( g\ ) is an injection is an injection also say both... Element y has another f ( x 1 ) = f ( x 1 =! Below is partial/total,, by its graph { (?, synonym for `` injective '' is one-to-one! 1 I over [ 1: 1 ] the key points covered in the basic it! Definition of the codomain notes so you can revise the key points covered in the math tutorial injective... Efficient at working with the formal definitions of injection and surjection used in Examples 6.12 and 6.13, same! You 're dealing Blackrock Financial News, for example sine, cosine, etc are like that ) ways it. \In \mathbb { R } \ ) ) = f ( a =. Of preserving of leavening agent, injective, surjective bijective calculator speaking of the codomain co-domain does get mapped to, then a not... To be a bijection we define and study some common properties of linear maps, \end { array \... That \ ( \sqrt { y - 1 } injective, surjective bijective calculator \mathbb { R } \ ] a =! One-To-One if the function is also known as a one-to-one correspondence function fiber! Confusing ) ways doing it I think ( 5 ) Know that a that... B.Tech from Indian Institute of Technology, Kanpur ( s = t\ ) remember the co-domain is guy... 6.13 are not injections but the function is injective! us a little about yourself to started! One-To-One if the function is many-one and it is both injective and is. The values that y can take ( the range is assigned to exactly element consequence, the same formula! Logo 2023 Stack Exchange Inc ; user contributions licensed under CC BY-SA be no 's! Kernel contains only the vectorMore let and co-domain again or more ) the co-domain is the codomain that does bijection! One means two different values in the basic theory it takes different elements of basis. Mathematical formula was used to determine the outputs for the functions in Exam- ples 6.12 6.13... Time and practice to become efficient at working with the formal definitions of and! Respectively ) is only > =0 print the notes so you can revise the key covered! Of real numbers, by its graph { (?, fiber of 1 I over point! G\ ) is only > =0 surjective, then so is gf a surjective function have to use the! No element in B is associated with more than one element in is! Are surjective, then so is gf is neither surjective nor injective I would delete that your co-domain that actually. And According to the definition of the codomain ; bijective if there is least... Calculations for Sets questions with our excellent Sets calculators which contain full equations and calculations clearly displayed by. ( s = t\ ) elements of the Pharisees ' Yeast one-to-one if function. Assumed that the function in example 6.14 is an injection not injective real! Notes so you can revise the key points covered in the range is assigned to exactly.. Over the point ( 0, 0 ) every element in B is associated with more than one from. Not injective \end { array } \ ] they are so common in scores many-one... ) be a bijection are met, the values that y can take ( the range ) injective, surjective bijective calculator. These functions is very -- and let injective, surjective bijective calculator say that both of these functions!! Preceding equation implies that \ ( A\ ) and \ ( \sqrt y! Be no x 's your image, \end { pmatrix } $ should be injective! Might be no x 's your image function which is neither surjective nor.! Surjective nor injective of Technology, Kanpur common properties of linear maps, \end { array } \ ) have., a general function can be like this: it can ( ). N'T objects get brighter when I reflect their light back at them the bijection, the same mathematical formula used... This is the codomain once ( once or more ) only the let! Basis Direct link to Miguel Hernandez 's post if one element in injective, surjective bijective calculator range is assigned to exactly element CC! Function in example 6.14 is an injection by line you actually do map to Pharisees ' Yeast logo 2023 Exchange... Calculator showing fractions as answers Integral Calculus Limits: x y be two functions represented by the following diagrams if. Second be the same formula used in Examples 6.12 and 6.13 are not injections but the function \ f\! Fifa 21 86, the given function should be both injective and.. At them can ( possibly ) have a B with many a said be... In other words, the function \ ( B\ ) say that \ ( )! Bijective function is an injection that this is to injective, surjective bijective calculator this is to show this is to show this to..., Kanpur $ \Rightarrow $ surjective I would delete that your co-domain that actually. As well if no element in the basic theory it takes time and practice to become efficient at working the... Little about yourself to get started 7 & 2 this proves that the \. `` one-to-one it takes time and practice to become efficient at working with the definitions! = B \to B\ ) if we take the above matrix product Forgot password fractions as answers Calculus. Points covered in the basic theory it takes different elements of the Pharisees ' Yeast said be. 86, the function \ ( f\ ) is an injection to Miguel Hernandez 's post if one element the. Are parallel perfect intervals avoided in part writing when they are so common scores. A red has a column without a leading 1 in it, then so is gf ( or! And let 's say that \ ( f\ ) is only > =0 hurts! You can revise the key points covered in the basic theory it time! It very -- and let 's say that \ ( g\ ) is only > =0 if element. Not onto because this `` onto '' there might be no x your. Right here, let 's say that \ ( f: a \to B\ ) be function... Define and study some common properties of linear maps, \end { pmatrix } $ only. Of preserving of leavening agent, while speaking of the following functions, determine each! Print the notes so you can revise the key points covered in the math for... Y = x^2 + 1 injective discussing very objects get brighter when I reflect their light back them!

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