Putting y = 3 Eg: Why does $g \circ f$ being injective imply that $f$ is injective too? This "hits" all of the positive reals, but misses zero and all of the negative reals. Can a map be subjective but still be bijective (or simply injective or surjective)? Therefore, $f$ is not surjective. x1 = x2 or x1 = x2 Since is surjective, there is an x2Q with (x) = 1. Stack Overflow at WeAreDevelopers World Congress in Berlin. Which lattice parameter should be used, the one obtained by vc-relax or the optimized value acquired through the Birch-Murnaghen equation? How to avoid conflict of interest when dating another employee in a matrix management company? Is it injective? Bijection, Injection, And Surjection | Brilliant Math & Science Wiki It is however true that the function $$g : [0,\infty)\to [0,\infty)$$$$g(h)=h^2$$ is bijective. In mathematics, a surjective function (also known as surjection, or onto function / n.tu /) is a function f such that every element y can be mapped from some element x such that f(x) = y. Prove that, if $f \circ f$ is injective, then $f$ is injective. After time you will get a feeling which one works the best to prove. But if I change the range and domain to $\operatorname{g}: \mathbb{R}^+ \to \mathbb{R}^+$ then it is both injective and surjective. An injective function should always be considered as not surjective. Is this mold/mildew? If mapping is surjective, then it's injective in finite sets. Then, choose $x=\sqrt{y-1}$, so that $f(x)=(\sqrt{y-1})^2+1=y-1+1=y$, and f is surjective. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Laplace beltrami eigenspaces of compact Lie groups. Hence the function is injective, since we proved that if any two elements map to the same output, they must. Since the contrapositive is logically equivalent, and in this case, is often much easier to prove, it suffices to prove: If $f(a) = f(b)$, then $a = b$, which is precisely the contrapositive and is the method being used. It only takes a minute to sign up. f (x2) = 1 + (x2)2 Okay, so "y" is simply another point, so if y = -1, that means that (-1)^2 = 1, so f(y) = 1. This implies a = b. Using get_feature function with attribute in QGIS. Definition.$\quad$ A mapping $\alpha\colon S\to T$ is said to be one-to-one or injective if $$x_1\neq x_2\quad\text{implies}\quad\alpha(x_1)\neq\alpha(x_2)\qquad(x_1,x_2\in S).$$. and caffeine. How about $f(x)=e^x.$ Your job is to figure out the domain and range. It only takes a minute to sign up. Stack Overflow at WeAreDevelopers World Congress in Berlin. Functions Solutions: 1. f is not one-one. Show f(x) = x2 is neither one-one nor onto - Examples - Teachoo Surjective Function How To Prove w/ 11+ Solved Examples! - Calcworkshop $$ Looking for story about robots replacing actors. Can you find a natural number that isn't a square of some natural number? Prove that the function $f(x) = x^2$ for $x\in \mathbb N$ is injective, but not surjective. The best answers are voted up and rise to the top, Not the answer you're looking for? Lets show that $f(x) = x^3$ is injective. An example of an injective function $\mathbb{R}\to\mathbb{R}$ that is not surjective is $\operatorname{h}(x)=\operatorname{e}^x$. Otherwise the function would be called a bijection. (Python), Chapter 1 Class 12 Relation and Functions. What you did is $\sqrt{y} = x$ but that not enough. Applying the third-root will give us $x=y$. f (x2) = (x2)2 Determining Injective, Surjective, Bijective Functions over range of Integers. Then 2(x=2) = (x) = 1, but there is no integer nwith 2n= 1. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. rev2023.7.25.43544. English abbreviation : they're or they're not. Surjective functions, also called onto functions, is when every element in the codomain is mapped to by at least one element in the domain. Therefore $f$ is injective. Answer (1 of 5): Depends on the choice of the domain and co-domain. Could ChatGPT etcetera undermine community by making statements less significant for us? It's the same as f(x1), f(x2). Prove that $f$ maps $ \mathbb R$ onto $[1, \infty)$. So using this information, how would I prove this problem? The meaning of "$y$" changes throughout the text you've written. Can a Rogue Inquisitive use their passive Insight with Insightful Fighting? How to Prove a Function is Not Surjective(Onto) - YouTube Proof. It never has one "A" pointing to more than one "B", so one-to-many is not OK in a function (so something like "f (x) = 7 or 9" is not allowed) But more than one "A" can point to the same "B" ( many-to-one is OK) . I am tempted to use the property $f(x) = y$ to replace $f(x)$ in $f(x) = x^2 + 1$ with $y$ and solve $y = x^2 + 1$ for $x$. Provide an example of each of the following. Note that D 12 has an element of order 12 (rotation by . The definition you had in class pretty much does the same. (Bathroom Shower Ceiling). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. For x = 2 x = 2, y = 4 y = 4. Take any bijective function $f:A \to B$ and then make $B$ "bigger". How does Genesis 22:17 "the stars of heavens"tie to Rev. This is a. You consistently write sentences where $f(2)=2^2$ is immediately followed by $f(4)=\sqrt4$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. x 12=x 22. Claim: ex e x is surjective. This is where I'm confused. minimalistic ext4 filesystem without journal and other advanced features. Surjective Injective Bijective Functions - Statistics How To By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Is it better to use swiss pass or rent a car? Therefore, x x and y y are not equal, so it's not injective. If $f$ where given as a function $f:\mathbb N \to A$ where $A=\{ n^2 \mid n \in \mathbb N \}$, then $f$. What would naval warfare look like if Dreadnaughts never came to be? Why isn't the e-power function surjective then? If you have two values like $x=-1$ and $y=1$ with property of $f(x) = f(y) = 1$ them $f$ cant be injective because two different values are mapping onto the same value. So using this information, how would I prove this problem? We take general $x,y \in \mathbb{R}$. (4) f: [0;1) ! Again, note that $y$ is unrelated to $x$. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo. It seems that you mean $f: \mathbb{N} \to \mathbb{N}$ given by $x \mapsto x^2$ is injective and not surjective. PDF Math 67A Homework 4 Solutions - UC Davis rev2023.7.25.43544. Do the subject and object have to agree in number? This illustrates the important fact that whether a function is surjective not only depends on the formula that defines the output of the function but also on the domain and codomain of the function. Will the fact that you traveled to Pakistan be a problem if you go to India? May I reveal my identity as an author during peer review? Here, f(1) = f(1) , but 1 1 For example: "Tigers (plural) are a wild animal (singular)". f(x 1)=f(x 2) x 12+2x 128= x 22+2x 228. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Proof by Contradiction to show that if $f^{-1}$ exists, $f$ must be onto. Are there any practical use cases for subtyping primitive types? Learn more about Stack Overflow the company, and our products. $f(x)=x^{3}+1$ - Injective and Surjective? May I reveal my identity as an author during peer review? x1 = x2 or x1 = x2 The other definition is just the other way around. (Or maybe tired.) Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. PDF Math 430 { Problem Set 4 Solutions - MIT Mathematics Some books, professors usually keep referring 'y' as the output of the function i.e. f(1) = 1 + (1)2 = 1 + 1 = 2 Thus (x) is prime. Please keep in mind that the graph is does not prove your conclusions, but may help you arrive at the correct conclusions . 6:13 when the stars fell to earth? However, according to the contrapositive, $x$ doesn't equal $y$ implies that $f(x)$ doesn't equal $f(y)$. Thank you for example $\operatorname{f} : \mathbb{R} \to \mathbb{C}$. Different balances between fullnode and bitcoin explorer. This is where you might messed up something. Its inverse function is called $\sqrt{\bullet}$. f(-2)=f(2)\:\text{ but }-2\:\text{ isn't equal to }2.$$, $$ B is bijective (a bijection) if it is both surjective and injective. Therefore, f f is injective. We must show that if y Y, then there exists an x such that f ( x) = y. I am tempted to use the property f ( x) = y to replace f ( x) in f ( x) = x 2 + 1 with y and solve y = x 2 + 1 for x. What is the practical benefit of a function being injective? In the circuit below, assume ideal op-amp, find Vout? How can kaiju exist in nature and not significantly alter civilization? Functions can be injections (one-to-one functions), surjections (onto functions) or bijections (both one-to-one and onto). What should I do after I found a coding mistake in my masters thesis? A few quick rules for identifying injective functions: If a function is defined by an odd power, it's injective. Airline refuses to issue proper receipt. Hence, it is injective. Putting f (x1) = f (x2) Right? Prove that S 4 is not isomorphic to D 12. Could it possibly be because it's not in the givens? https://goo.gl/JQ8NysHow to Prove a Function is Not Surjective(Onto) A function is injective if no two inputs have the same output. In general, you may want to use the fact that strictly monotone functions are injective. For all x X, there exists a unique y Y such that f(x) = y . Now it is still injective but fails to be surjective. (Python), Class 12 Computer Science To be precise, the exact range is all polynomials whose coefficient of degree $0$ and $1$ are zero. (A modification to) Jon Prez Laraudogoitas "Beautiful Supertask" What assumptions of Noether's theorem fail? Please login :). I realize that $y=x^2$ is not injective. English abbreviation : they're or they're not. So, let's suppose that f(a) = f(b). Examine if the function is injective, how to interpret the result of proof. Then $x^2=y \Rightarrow x^2 = 17 \Rightarrow x = \pm\sqrt{17} \notin \mathbb{N}.$ You have found a $y$ such that $x \notin \mathbb{N}$ for $f(x)=y$. Prove/Disprove $f(x)=e^{x}$ is Injective and Surjective Again, following that helpful answer, solve for x x, and then plug into f(x) f ( x): Now, plug x x into f f, and check if result equals y y, which would prove the surjective property. Why is a dedicated compresser more efficient than using bleed air to pressurize the cabin? If x < 0 x < 0, ex = 1 ex e x = 1 e x, hence also positive. To help Teachoo create more content, and view the ad-free version of Teachooo please purchase Teachoo Black subscription. Connect and share knowledge within a single location that is structured and easy to search. Here, the set $S$ is the domain and $T$ is the codomain. x_1=x_2. The function f:R R , f(x) = x^2 is - Toppr Why does ksh93 not support %T format specifier of its built-in printf in AIX? Question: Prove that the function g:R to R* defined by g (x) = 2x is an injective homomorphism that is not surjective. I just solved for for $x$ when $y=f(x)$. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ; Surjective: is there $x\in\mathbb N$ such that $x^2=2$? Surjective means that every "B" has at least one matching "A" So B is range and A is domain. Please Subscribe here, thank you!!! This problem has been solved! It is an upward parabola with no real root. Show that the mapF: R2R2 given byF(x, y)=(x+y, x+ 1) is not linear. It is not one-one Example 11 Show that the function f: R R, defined as f (x) = x2, is neither one-one nor onto f (x) = x2 Checking one-one f (x1) = (x1)2 f (x2) = (x2)2 Putting f (x1) = f (x2) (x1)2 = (x2)2 x1 = x2 or x1 = -x2 Since x1 does not have unique image, It is not one-one Eg: f (-1) = (-1)2 = 1 f (1) = (1)2 = 1 Here, f (-1) = f . The statement in class is correct, and you example of $x=1, y=-1$ proves the function is not injective because you have $f(x)=f(y)$ but $x \neq y$. From there you would like to show that $a$ must be equal to be $b$ in order to satisfy the "uniqueness" property of injective functions. 8.2: Injective and Surjective Functions - Mathematics LibreTexts Airline refuses to issue proper receipt. Hence, it is not one-one (Python), Class 12 Computer Science Is this mold/mildew? Assume that X and Y are finite sets. How to avoid conflict of interest when dating another employee in a matrix management company? Since gis injective, we have Injective 2. PDF Functions Surjective/Injective/Bijective - University of Limerick Injective, Surjective and Bijective Functions - Online Tutorials Library Every function is surjective onto its image but this does not help with many problems. Airline refuses to issue proper receipt. This means that the general unknown $x,y$ you have picked are actually the same. For all y range(f), there is a unique x X such that y = f(x) . I learned about terms like surjective, injective and bijective so long ago, it seems like these terms aren't so popular anymore. And I have no idea how you bring $\mathbb{Z}_+$ into this. Injective Bijective Function Denition : A function f: A ! You are missing all polynomials of degree 0 0 and 1 1. If so, why would it be wrong to include that bit of info? Example2.2.1.LetA= fa; x2 = y 1 Let $y \in [1,\infty]$. $$ Here, 2 x - 3 = y So, x = ( y + 5) / 3 which belongs to R and f ( x) = y. surjective? I guess that makes sense. Is not listing papers published in predatory journals considered dishonest? So, f is not onto. As @N.F.Taussig noted, your definition of $f$ is technically incomplete. Let me take an example. Example 4.3.9 Suppose A and B are sets with A . Does the US have a duty to negotiate the release of detained US citizens in the DPRK? How to prove that function ${f(x) = x \oplus T}$ is injective or surjective? Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class, Example 11 Can I spin 3753 Cruithne and keep it spinning? If f ( x 1) = f ( x 2), then 2 x 1 - 3 = 2 x 2 - 3 and it implies that x 1 = x 2. For example $\sqrt{y = 1} = \pm 1$ which makes perfectly sense because both $x = -1$ and $x = 1$ are mapping onto $y = 1$. An example of an injective function $\mathbb{R}\to\mathbb{R}$ that is not surjective is $\operatorname{h}(x)=\operatorname{e}^x$. You just need to give a counter-example here. y=f(x), the op may have become confused with that. Solved Prove that the function g:R to R* defined by g(x) - Chegg Thus cannot exist. 6.3: Injections, Surjections, and Bijections - Mathematics LibreTexts (-2 \le y \le 10\). What would naval warfare look like if Dreadnaughts never came to be? Suppose $f:S \rightarrow S$ for some set $S$. Since x1 does not have unique image, Connect and share knowledge within a single location that is structured and easy to search. In the range of T T you only have polynomials of degree 2 2 and the zero polynomial p(x) 0 p ( x) 0. Do both the contrapositive and the contrapositive of the contrapositive have to be true for it to be injective? 10x 12=10x 22. Problem 8.28. That means that $f(x) = x^3$ is injective. Dividing both sides by 2 gives us a = b. What information can you get with only a private IP address? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. To prove that a function is not injective, we demonstrate two explicit elements and show that . PDF 447 HOMEWORK SET 1 - University of Tennessee (Bathroom Shower Ceiling). Note that the $\implies$ there could be replaced by $\iff$, since clearly $a=b\implies f(a)=f(b)$. Find the matrix forTwith respect to the canonical basis of R2. How can kaiju exist in nature and not significantly alter civilization? Should I trigger a chargeback? The best answers are voted up and rise to the top, Not the answer you're looking for? Prove that f maps R onto [ 1, ). Do the subject and object have to agree in number? You also get $\sqrt{y} = - x$. 1 + x2 = y Does glide ratio improve with increase in scale? For them we say that $f(x) = f(y)$. What would naval warfare look like if Dreadnaughts never came to be? Injective function: example of injective function that is not surjective. Let x 1 and x 2 be two elements in the domain (R), such that. Example "/\v[\w]+" cannot match every word in Vim. Different balances between fullnode and bitcoin explorer. Prove/Disprove $f(x)=e^{x}$ is Injective and Surjective, Proof showing $f(x)$ is injective confusion, Prove $f(x) \in f(A) \implies x \in A$ if $f$ is injective and $b \in B \implies f^{-1}(b) \in f^{-1}(B)$ if $f$ is surjective, Injective, surjective and bijective functions. When we change the image to $ \mathbb{C} $ in the first example, how should we constrain it to make it surjective? How difficult was it to spoof the sender of a telegram in 1890-1920's in USA? We can see this because the equation $x^2=1$ has two solutions, so even though $g(-1)=1=g(1)$ the inner parts are not equal. Connect and share knowledge within a single location that is structured and easy to search. Proving f ( x) = x 2 + 1 is surjective. Let $f(x) = x^2 + 1$, where $x$ is a real number. EXAM 2 SOLUTIONS Problem 1.IfRis an equivalence relation on a nite nonempty setA, then the equivalence classes of all have the same number of elements. Putting f(x1) = f(x2) This means that $f$ cant be injective. But then I can change the image and say that $\operatorname{f} : \mathbb{R} \to \mathbb{C}$ is given by $\operatorname{f}(x) = x^3$. Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. Share Cite Follow The best answers are voted up and rise to the top, Not the answer you're looking for? Was the release of "Barbie" intentionally coordinated to be on the same day as "Oppenheimer"? Putting y = 3 By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. It just all depends on how your define the range and domain. how can i prove if f (x)= x^3, where the domain and the codomain are both the set of all integers: Z, is surjective or otherwise.the thing is, when i do the prove it comes out to be surjective but my teacher said that it isn't. this is what i did: y=x^3 and i said that that y belongs to Z and x^3 belong to Z so it is surjective Solution. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Teachoo gives you a better experience when you're logged in. We use it with inverses and transcendental functions in Calc. Transcript. multiplication by $x^2$, defined by $T \in \mathbb{L(P(R),P(R))}$ by $$(Tp)(x) = x^2p(x)$$. Therefore, fis injective and surjective, and thus, bijective. Proof. Was the release of "Barbie" intentionally coordinated to be on the same day as "Oppenheimer"? -1 Prove that the function f ( x) = x 2 for x N is injective, but not surjective. Displaying ads are our only source of revenue. How to get the chapter letter (not the number). 3 Answers Sorted by: 4 Let f: R R, x 1 x2 f: R R, x 1 x 2. Hence, it is not one-one Show that the function f: R R, defined as f(x) = x2, is neither one-one nor onto Can a creature that "loses indestructible until end of turn" gain indestructible later that turn? How do you prove that a function is surjective? - Physics Forums You'll get a detailed solution from a subject matter expert that helps you learn core concepts. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Would that be permissible what with the property $f(x) = y$ being in the conclusion of the statement, rather than in the hypothesis and given conditions? Can a creature that "loses indestructible until end of turn" gain indestructible later that turn? To help Teachoo create more content, and view the ad-free version of Teachooo please purchase Teachoo Black subscription. Since $f(x)=2x$, he considers the following, where $x_1$ and $x_2$ are presumed to be elements in $\mathbb{Z}$ (this is why it's important to specify your domain and codomain; otherwise, it's ambiguous): If a function is defined by an even power, it's not injective. Cardinality, surjective, injective function of complex variable. https://goo.gl/JQ8NysHow to Prove a Function is Surjective(Onto) Using the Definition Let's look at that more closely: A General Function points from each member of "A" to a member of "B". A car dealership sent a 8300 form after I paid $10k in cash for a car. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. x = (1) x=y. Let f(x) be the height of person x, to the nearest inch. It is not one-to-one ($1$ and $-1$ both map to 1, for example). What is the definition of surjective according to you? Like $A \Rightarrow B$ is equal to $\neg B \Rightarrow \neg A$. Was the release of "Barbie" intentionally coordinated to be on the same day as "Oppenheimer"? Find the matrix forT with respect to the canonical basis for the domainR2 ((1,1),(1, 1)) for the target spaceR2. Checking one-one Functions $\mathbb{N} \to \mathbb{N}$ that are injective but not surjective, and vice versa, Construct a function that is surjective, but not injective. For example: "Tigers (plural) are a wild animal (singular)", How to get the chapter letter (not the number). Departing colleague attacked me in farewell email, what can I do? There are many examples. He has been teaching from the past 13 years. x = ((3)) Thus, if PQ (x) then the product of their constant terms is 0, and since Z is an integral domain, this means one of them has a constant term equal to 0, hence lies in (x). Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class, Ex 1.2, 7 rev2023.7.25.43544. (Bathroom Shower Ceiling). Presumably, your author is considering the mapping $f\colon\mathbb{Z}\to\mathbb{Z}$ defined by $f(x)=2x$, but you need to make sure the domain and codomain are clear at the outset. $\Bbb P(\Bbb R)$ stands for the set of polynomials with real coefficients, and $x^2$ is treated as a polynomial. Which is not possible as root of negative number is not real Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. functions - Understanding why $f(x)=2x$ is injective - Mathematics However, why wouldn't 1 be in range$(T)$? (ii) f: R R defined by f(x) = 1 + x2 For example, the following function is not injective on $[-1,1]$, $g(x) = x^2$. f(x) = x2 It is not one-one Equivalent to this is to prove that $f(a)=f(b)\Rightarrow a=b$, which is done there. How do you manage the impact of deep immersion in RPGs on players' real-life? Can a Rogue Inquisitive use their passive Insight with Insightful Fighting? Learn more about Stack Overflow the company, and our products. What is the most accurate way to map 6-bit VGA palette to 8-bit? However, according to the contrapositive, x x doesn't equal y y implies that f(x) f ( x) doesn't equal f(y) f ( y). Was the release of "Barbie" intentionally coordinated to be on the same day as "Oppenheimer"? Proof. Therefore f is injective. $$ Consequently, the "$y$" in "$f(y)$" is just some dummy variable that gets input into $f$ and is unrelated to $x$. Answer (1 of 6): Is it injective? Determining whether the following is injective, surjective, bijective, or neither. It is likely that $\Bbb {P(R)}$ denotes the set of polynomials with real coefficients. math.stackexchange.com/questions/991894/, Stack Overflow at WeAreDevelopers World Congress in Berlin. f(x) = 1 + x2 For $x = 2$, $y = 4$. ie: If $a \neq b$, then $f(a)\neq f(b)$. Solving the equation for $y$ in terms of $x$ is a perfectly valid (and in most cases most direct) way to do that. x\neq y\:\text{ but }\:f(x)=f(y)=4. In your problem, you have the mapping $f$ defined by $f(x)=2x$, but what are the domain and codomain? To prove that a function is injective, we start by: "fix any with " Then (using algebraic manipulation etc) we show that . Suppose that : Q !Z is an isomorphism. Please login :). Teachoo answers all your questions if you are a Black user! Find an infinite set $S$ and a function $g : S \to S$ that is surjective but not injective. This means that for all "bs" in the codomain there exists some "a" in the domain such that a maps to that b (i.e., f (a) = b). Prove that Q is not isomorphic to Z. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. @imranfat It depends completely on the range and domain. It only takes a minute to sign up. A function is abijectionif it is both injective and surjective. and caffeine. . Learn more about Stack Overflow the company, and our products. But you wouldn't write down that you solve for $x$ when $y = f(x)$ in the proof since that would be wrong, correct? This "hits" all of the positive reals, but misses zero and all of the negative reals. Can a Rogue Inquisitive use their passive Insight with Insightful Fighting? x2+1 for x, in terms of y. The goal here is to start by supposing that $f(a)$ and $f(b)$ take the same $y$ value. Without those, the words "surjective" and "injective" have no meaning. For example, $f(1) = f(-1)$. Ex 1.2, 2 (i) - Check the injectivity and surjectivity of f: N N Do the subject and object have to agree in number? Intuitively, I understand why $$f(x)=2x$$ is injective, but I don't understand the above proof. Problem Prove that a function f: R R defined by f ( x) = 2 x - 3 is a bijective function. Informally, an injection has each output mapped to by at most one input, a surjection includes the entire possible range in the output, and a bijection has both conditions be true. In the range of $T$ you only have polynomials of degree $\geq 2$ and the zero polynomial $p(x)\equiv 0$. y=(\text{expression of }x). If you want my opinion, you'll have a hard time reconciling what your book says with whatever you're trying to say, since your book and you use two different notations.