Invertibility as a two-sided undo, and why bijective is exactly the condition

What to assume they know

• Feeding one function into another, and why swapping them changes the answer — composition, the chain condition, and reading the composite notation
• Bijections, and why on a finite set either half implies the other — bijections, and the two demands checked separately
• The identity function, which returns each input unchanged
• Solving a linear equation for the unknown
• Parity of a natural number
• What it means for two functions to be equal

What they should be able to do

• State Definition 9, naming both equations and which set each identity function lives on
• Explain why the input-side equation forces the function to be one-one, and the output-side equation forces it to be onto
• Construct the inverse of a bijection target by target, and say why the construction defines a function
• Prove a function invertible by proving it one-one and onto, without producing the inverse
• Produce the inverse of a stated function and verify both equations
• Give a function with a one-sided undo that is not invertible, and say which equation fails
• Explain why a co-domain declared to be the range makes the output-side equation attainable
• Recognise a function that is its own inverse, and read the inverse notation correctly

Where it usually goes wrong

• "One equation is enough; the other follows." It does not. The doubling function on the naturals satisfies the input-side equation with a suitable g and fails the output-side one; the folding function does the reverse. Both are in this chapter, one page apart, and neither is invertible.
• "Invertible means you can rearrange the formula for x." Rearranging works when it works, and it is not the definition. Example 17's rearrangement produces a formula that is not even defined on all of N; the declared co-domain is what rescues it. The definition is about two composites equalling two identities, and a formula is only evidence.
• "The superscript minus one means one divided by the function." It does not. For the reciprocal function on the non-zero reals the two happen to agree, which is exactly the coincidence that entrenches the error. For the function sending x to 3 – 4x, the inverse sends y to the quotient of 3 – y by 4, and one divided by the function is a different thing entirely.
• "You have to find the inverse to prove a function invertible." The chapter says the opposite in the sentence right after Definition 9: proving one-one and onto is enough. That is the practical payoff of the whole topic.
• "Proving one-one and onto is easier than finding the inverse, so it is a shortcut." It is not a shortcut past anything; it is the same content differently arranged. The backward proof of section 6 constructs the inverse from the two properties, so nothing is skipped — you simply never write the formula.
• "If a function has an undo on paper, it is invertible." Ask which side the undo works on. The two worked one-sided constructions above are the diagnostic.
• "The two identity functions in Definition 9 are the same function." They live on different sets. When the two sets differ, as in Example 17, the two identities are visibly different objects.

Questions to check understanding

• Show that a stated function is invertible and find its inverse, verifying both composites — the form of Example 17
• Show that a stated function is invertible by proving it one-one and onto, without computing the inverse
• Given a function and a candidate inverse, verify or refute each of the two equations separately
• Explain why a stated function is not invertible, naming which of the two demands fails
• Show that a stated function is its own inverse
• Justify a declared co-domain by showing the function fails to be onto without it

Examples worth working on the board

Values marked verified are worked out here on the chapter's own data; no answer key was consulted, and the chapter prints no answers to its exercises.

• Definition 9 (§1.4, Part I p. 12). A function from X to Y earns the name when some function from Y back to X composes with it, in both orders, to give the identity — on X in one order and on Y in the other. That second function is then called the inverse and is written with the superscript minus one. Two equations, two different identity functions, two different sets. Sections 1 to 3 exist to stop the student reading this as one equation written twice.
• The chapter's claim, and its status (Part I p. 12). The paragraph immediately after Definition 9 asserts both directions at once — invertible gives one-one and onto, and one-one and onto gives invertible — and adds that this is what lets you establish invertibility without computing the inverse. No proof is printed for either direction. This was checked on the printed page, not only in the extracted text. Sections 4 to 6 supply the argument.
• The forward proof to supply (not in the book). Suppose some g satisfies the input-side equation. If two inputs share an image, apply g to both sides of that equality: each side collapses to its own input, so the two inputs were equal. Hence the function is one-one. Now suppose g satisfies the output-side equation. Given any target y, the element g(y) is an input whose image is y, so the target is reached. Hence the function is onto. Each equation buys exactly one property, and neither buys the other — which is the entire thesis in two paragraphs.
• The backward proof to supply (not in the book). Suppose the function is one-one and onto. Take any target y. Onto says at least one input maps to y; one-one says at most one does; so exactly one does, and sending y to that unique input is a well-defined rule from Y to X. Composing the two in either order returns the element you started with, by construction. Note where each half was spent: onto gave existence, one-one gave uniqueness, and a rule needs both before it is a function at all.
• Uniqueness of the inverse (not in the book, optional). If two functions both undo the original on both sides, compose one with the original and then with the other; associativity of composition collapses the expression to each of them in turn, so they are equal. One line, and it justifies saying the inverse.
• Example 17 (Part I pp. 12–13). The function from N to Y sends x to 4x + 3, where Y is declared as the set of naturals of that very form. The chapter takes an arbitrary element of Y, uses the definition of Y to recover x as the quotient of y – 3 by 4, defines the candidate inverse by that formula, and then verifies both equations, obtaining the identity on N in one order and the identity on Y in the other. Verified: Y is the set 7, 11, 15, 19, and so on. The candidate inverse lands in N precisely for those y, which is why the declared Y is not decoration — the formula for the inverse is not a function on all of N. The chapter has manufactured a co-domain equal to the range, which is what makes the output-side equation attainable, and section 9 is built on noticing that.
• A one-sided undo, input side (Example 8, Part I p. 8; the construction is added here). The function from N to N sending x to 2x is one-one and not onto. Define g from N to N by halving every even number and sending every odd number to 1. Verified: composing g after the doubling returns every natural unchanged, so the input-side equation holds. But composing the doubling after g sends 1 to 2, not to 1, so the output-side equation fails. The function has an undo on one side only and is not invertible. This is the sharpest possible demonstration that the two equations of Definition 9 are independent demands.
• A one-sided undo, output side (Example 10, Part I p. 9; the construction is added here). The function from N to N sending both 1 and 2 to 1, and every x above 2 to x – 1, is onto and not one-one. Define g from N to N by sending 1 to 1 and every y at least 2 to y + 1. Verified: composing the original after g returns every natural unchanged, so the output-side equation holds — for y at least 2 the input y + 1 exceeds 2 and maps to y, and 1 maps to 1. But composing g after the original sends 2 to 1, not to 2, so the input-side equation fails. Put beside the previous item this is a matched pair: one function with each half of the definition and neither of them invertible.
• A function that is its own inverse (Example 12, Part I pp. 9–10; the observation is added here). The function from N to N sending an odd x to x + 1 and an even x to x – 1 meets both demands, so it is invertible. Verified: composing it with itself returns every input unchanged — an odd x rises to the even x + 1 and comes back down; an even x drops to the odd x – 1 and comes back up. So the inverse is the function itself. It is the easiest inverse in the chapter and the chapter never mentions it.
• A second self-inverse function (Exercise 1.2 Q1, Part I p. 10; the observation is added here). The reciprocal on the non-zero reals is one-one and onto, so it is invertible, and applying it twice returns the input. Verified: the reciprocal of the reciprocal of a non-zero x is x.
• Inverses to compute, all bijections established elsewhere in the chapter (Exercise 1.2, Part I p. 11; the inverse formulas are working added here). Q7(i): the function on R sending x to 3 – 4x is a bijection, and its inverse sends y to the quotient of 3 – y by 4. Q10: the function from the reals without 3 to the reals without 1 sending x to the quotient of x – 2 by x – 3 is a bijection, and its inverse sends y to the quotient of 3y – 2 by y – 1. Miscellaneous Exercise Q1 (Part I p. 15): the function from R to the reals strictly between –1 and 1 sending x to the quotient of x by one plus the size of x is a bijection, and its inverse sends y to the quotient of y by one minus the size of y. Verified by substituting each inverse into the original in both orders. Use one of these for section 10 and hold the other two as practice.
• The identity function's own definition (Miscellaneous Example 25, Part I p. 14). The chapter writes the identity function on N explicitly, sending each input to itself. Note the sequencing: Definition 9 on p. 12 uses the identity notation two pages before this, and Class XI supplies the meaning in between.

Figures to have open

• A two-lane diagram of the two equations: one lane starting in X and returning to X, the other starting in Y and returning to Y, each labelled with its own identity. This is an added device and carries sections 1 to 3, 7 and 8. It is the most important picture in the topic.
• An arrow diagram of the doubling function on the first six or eight naturals with its one-sided undo drawn back, and the single element where the reverse composite goes wrong picked out. Not in the book; the chapter states the function and draws nothing.
• The matching diagram for the folding function. Not in the book.
• A redraw of Fig 1.5 (Part I p. 12) may be reused from Feeding one function into another, and why swapping them changes the answer for section 1, since both equations are composites.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 1 "Relations and Functions", §1.4 Composition of Functions and Invertible Function, Definition 9 and the paragraph following it, p. 12
• Example 17, pp. 12–13
• Examples 8 and 10, pp. 8–9, and Example 12, pp. 9–10, for the functions used in sections 7, 8 and 11
• Exercise 1.2, questions 1, 7 and 10, pp. 10–11, and Miscellaneous Exercise question 1, p. 15, for the bijections whose inverses are computed
• Miscellaneous Example 25, p. 14, for the identity function written out