Feeding one function into another, and why swapping them changes the answer

What to assume they know

• What one-one asks of distinct inputs, and what many-one allows and Onto as the demand that nothing in the codomain be left unhit — one-one and onto, and the range of a function
• Domain, co-domain and range from Class XI
• Evaluating a function at a stated input, including at zero
• The cosine of an angle, and its value at zero
• Squaring, and multiplying a square by a constant
• What has to be shown to establish that two functions are not equal

What they should be able to do

• State Definition 8 and identify the three sets involved and which two must match
• Compute the composite of two functions given on small finite sets, value by value
• Read the composite notation correctly, naming which function acts first
• Decide whether the reversed composite is defined at all, from the declared sets alone
• Compute both composites for two functions on the reals and show they are unequal by evaluating at one input
• Explain why agreement at some inputs does not make two functions equal
• Describe what a composite inherits from its two factors, and what it does not

Where it usually goes wrong

• "The composite written with the second letter first means the second function acts first." It is the other way round. The letter nearest the input in the written value rule is the one that touches the input, and that is the inner letter. Every arithmetic slip in this section traces back to this.
• "If both functions are defined, both composites are defined." Example 15 refutes it directly: the second function's target set and the first function's source set are different sets, so the reverse composite cannot even be written down. Definition 8 is stated with the chain of three sets for exactly this reason.
• "The two composites are equal if the functions are nice enough." Cosine and three times a square are as nice as functions get and their composites differ at the very first input anyone would test.
• "To show two functions are unequal I must show they differ everywhere." One input is enough, and the chapter uses exactly one. Conversely, to show they are equal you must handle every input, which is why the equality of functions is the harder claim.
• "The two composites agreed at the input I tried, so they are the same function." Agreement at a point is a coincidence, not an argument. Ask what would be needed to conclude equality, and the answer is every point of the common domain.
• "Composing two one-one functions can give a many-one one." It cannot, and the composite in Example 15 is many-one only because its factors are. Distinguishing what a composite inherits from what it invents is worth doing once, carefully, on the chapter's own numbers.
• "The composite's co-domain is the middle set." It is the last set. Fig 1.5 draws the composite's arc from the first oval to the third precisely to make this visible.

Questions to check understanding

• Find the composite of two functions given on small finite sets, listing its values
• Find both composites of two functions on the reals and show they are unequal by evaluating at a chosen input
• State whether a stated composite is defined, and if not, say which two sets fail to match
• Compute the value of a composite at a named input
• Show that a stated function composed with itself returns every input unchanged

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 8 and Fig 1.5 (§1.4, Part I p. 12). Three sets A, B and C, with the first function carrying A to B and the second carrying B to C; the composite is declared as a function from A to C, and its value at an input is obtained by applying the second function to the first function's output. Read off the page image, the figure draws three ovals labelled A, B and C left to right, each holding one marked dot — the input in A, its image in B, and that element's image in C — with the two arrows across the top labelled with the two function names, and a long arc beneath the whole figure labelled with the composite.
• Example 15 (Part I p. 12). The first function runs from {2, 3, 4, 5} to {3, 4, 5, 9} and sends 2 to 3, 3 to 4, and both 4 and 5 to 5. The second runs from {3, 4, 5, 9} to {7, 11, 15} and sends both 3 and 4 to 7, and both 5 and 9 to 11. The composite is asked for. Verified: 2 goes to 3 and then to 7; 3 goes to 4 and then to 7; 4 goes to 5 and then to 11; 5 goes to 5 and then to 11. So the composite sends 2 and 3 to 7, and 4 and 5 to 11. Two further facts, both the explanation's, are worth ten seconds each: the composite is many-one, and 15 in the final target is reached by nothing — indeed the second function never produces 15 in the first place. And the reverse composite is not defined at all, because the second function's target set {7, 11, 15} is not the first function's source set. That is section 6 and it is the strongest point in the topic.
• Example 16 (Part I p. 12). Both functions run from R to R; one sends x to the cosine of x, the other sends x to three times x squared. Both composites are asked for, and the chapter concludes they are unequal. Verified: feeding cosine into the squaring function gives three times the square of the cosine of x; feeding the squaring function into cosine gives the cosine of three x squared. At the input 0 the first is three times the square of 1, that is 3, and the second is the cosine of 0, that is 1. Since 3 and 1 differ, the two composites are different functions. One input was enough — that is the whole method of section 8.
• The order of writing against the order of acting (Definition 8, Part I p. 12). The composite's name is written with the outer function's letter first and the inner function's letter second, while the inner function is the one that acts first. The chapter states the value rule and does not comment on the reversal, and it is the single most reliable source of error in this section. Section 2 should say it out loud and show it.
• What a composite inherits (not in the book, worked on Example 15's data). Collisions survive composition: 4 and 5 already collide under the first function, so they collide under the composite. New collisions can also appear: 2 and 3 do not collide under the first function, and they do under the composite, because the second function folds 3 and 4 together. Missed targets accumulate the same way. The chapter computes the composite and stops; this is an added reading of the numbers it computes, and it should be labelled as such.
• A composite that recovers the input (Example 17, Part I pp. 12–13). The function from N to Y sending x to 4x + 3, composed with the function sending y to the quotient of y – 3 by 4, returns each input unchanged, and the two composed the other way returns each target unchanged. Use it only as the closing pointer of section 11; it belongs properly to Invertibility as a two-sided undo, and why bijective is exactly the condition.
• A composite of a function with itself (Example 12, Part I pp. 9–10; the composition is added here). The function from N to N sending an odd x to x + 1 and an even x to x – 1, composed with itself, returns every input unchanged. Verified: an odd x goes to the even number x + 1 and then back to x; an even x goes to the odd number x – 1 and then back to x. This is the chapter's own function and it makes section 11's point without any new machinery.

Figures to have open

• A redraw of Fig 1.5 (Part I p. 12): three ovals in a row, each with one marked element, two labelled arrows across the top and one labelled arc beneath. From the textbook's figure, redrawn. The arc must start at the first oval and end at the third.
• A two-box series diagram whose connector visibly fits or does not fit, for sections 1 and 6. An added device and the carrier of the whole chain-condition argument.
• An arrow diagram of Example 15's two functions drawn as one three-column picture, for sections 4 and 5. The chapter gives the values in prose and draws nothing.

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 8 and Fig 1.5, p. 12
• Examples 15 and 16, p. 12
• Example 12, pp. 9–10, for the function used as the self-composition specimen
• Example 17, pp. 12–13, as the forward pointer only