The one-output rule that promotes a relation to a function

What to assume they know

• A relation is nothing more than a chosen part of the product — a relation as a chosen subset of a product
• What goes in, what comes out, and what was merely allowed to come out — domain, range and codomain, and why range sits inside codomain
• Substituting into a linear or quadratic expression and evaluating it
• Reading a piecewise description: which formula applies on which stretch of inputs
• The natural numbers, the integers, and the even numbers

What they should be able to do

• Separate Definition 5 into its two demands and state each one as a test on a list of pairs
• Decide whether a listed relation is a function, and when it is not, say which of the two demands failed
• Explain why repeated closing entries do not disqualify a relation, and produce an example where many inputs share one output
• Use the arrow notation for a function, and name what the image and the preimage of a stated element are
• Evaluate a function at a list of inputs and complete a table, as Example 12 does
• Apply Definition 6 to decide whether a given function is real valued, and whether it is a real function
• Test a piecewise relation for functionhood by checking the shared input where its pieces meet
• Justify, from the definition, why a stated set of pairs fails to be a function from one named set to another

Where it usually goes wrong

• "Two inputs must not share an output." That is the wrong direction and it is the commonest error here. Exercise 2.3 Q1(i) sends six inputs to the single number 1 and is a perfectly good function. The forbidden thing is one input with two outputs.
• "If there is a formula, it is a function." Miscellaneous Exercise Q1's second relation is given entirely by formulas and still fails, because those formulas disagree at the input they share.
• "Example 7 fails because 6 is not an output." It fails because 6 is not an input. In fact 6 is an output there, as the image of 5. Get the direction right.
• "A function must use every element of the second set." That was settled in the previous topic: the range may sit strictly inside the codomain, and Example 10's range is only the even natural numbers.
• "A function is a rule." Definition 5 is about a relation, which is a set of pairs. The rule is one way of specifying which pairs.
• "Real valued and real function mean the same thing." Definition 6 imposes a condition on the outputs first and then a second condition on the inputs, so the second name is strictly the narrower of the two. But do not reach for a function on the natural numbers to separate them: the naturals are themselves a set of real numbers, so Example 12 satisfies both conditions and is a real function as well as a real valued one. The conditions come apart only when the inputs are not numbers at all — the letters-to-names relation of §2.3, p. 28, is the chapter's own case of that.
• "Piecewise means two different functions." Miscellaneous Exercise Q1's first relation is one function, defined by two clauses that happen to agree where they overlap. Whether it is one function is exactly the question being asked.

Questions to check understanding

• Decide whether a listed set of pairs is a function and give the reason, in the form Exercise 2.3 Q1 uses
• Give the domain and range of a relation once it has been shown to be a function
• Evaluate a function at several stated inputs, including a negative one
• Run a function backwards: given the output, find the input, as Exercise 2.3 Q4 part (iv) does
• Show that one piecewise relation is a function and a second, similarly built, is not
• Justify whether a stated collection of pairs is a relation from one set to another, and separately whether it is a function
• Explain in words why an element appearing twice as a first entry is fatal while an element appearing twice as a second entry is not

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data.

• Example 7 revisited (p. 31). A holds 1 through 6, and the relation keeps the pairs whose closing entry is one more than the opening. Verified: the five pairs never repeat an opening entry, so the single-output demand is met, and the relation still fails because 6 opens no pair at all. The chapter's own wording is that 6 has no image. Show the five arrows and the one bare dot.
• Example 8 revisited (p. 31). The squares relation of Fig 2.6. Verified: 9 is sent to both 3 and −3, so the second demand fails, while every element of the left oval does open a pair. The two failure modes are cleanly separated across these two examples.
• The pointer at Example 9 (p. 31). The chapter also says the relation of Example 9 is not a function and leaves the reason as a question. Example 9's stated task is to count relations, and the object written out there is the full product of the two-element sets. Taken as a relation, that product sends 1 to both 3 and 4, which is the second failure again. Verified: the product has four pairs and two of them open with 1.
• Example 11 (p. 31). Three rosters to test. (i) 2 with 1, 3 with 1, 4 with 2. Verified: openings 2, 3, 4 all distinct, so a function; domain 2, 3, 4; range 1 and 2. Note that 1 is the image of two different inputs and this is permitted. (ii) 2 with 2, 2 with 4, 3 with 3, 4 with 4. Verified: 2 opens twice, so not a function. (iii) 1 with 2, 2 with 3, 3 with 4, 4 with 5, 5 with 6, 6 with 7. Verified: six distinct openings, so a function; domain 1 to 6, range 2 to 7.
• Example 10 (p. 31). On the natural numbers, the closing entry is twice the opening. Verified: every natural number has exactly one double, so this is a function; the domain and the codomain are both the natural numbers, and the range is the even ones.
• Example 12 (p. 31). A function on the natural numbers doubling the input and adding one, tabulated at 1 through 7. Verified: the outputs are 3, 5, 7, 9, 11, 13, 15.
• Miscellaneous Exercise Q1 (p. 40). Two piecewise relations, and the sharpest item in the chapter. The first squares its input on the stretch from 0 to 3 and triples it on the stretch from 3 to 10. The second squares its input from 0 to 2 and triples it from 2 to 10. Verified: the two stretches overlap at exactly one input in each case. In the first, 3 squared is 9 and three times 3 is also 9, so the two clauses agree and only one output is produced — a function. In the second, 2 squared is 4 while three times 2 is 6, so the input 2 is sent to two different numbers — not a function. Nothing else distinguishes the two definitions, which is what makes the item worth a whole section.
• Miscellaneous Exercise Q10 (p. 40). A holds 1, 2, 3, 4; B holds 1, 5, 9, 11, 15, 16; the pairs are 1 with 5, 2 with 9, 3 with 1, 4 with 5, 2 with 11. Verified: every pair lies inside the product, so A and B do stand in a relation here; but 2 opens twice, with 9 and with 11, so it is not a function. Note also that 4 and 1 share the image 5, which is not a fault.
• Miscellaneous Exercise Q11 (p. 41). The pairs whose opening is a product of two integers and whose closing is their sum. Verified: 6 arises as 2 times 3 and as 1 times 6, giving the sums 5 and 7, so 6 opens two pairs with different closings and this is not a function from the integers to the integers.
• Exercise 2.3 Q1 (p. 38). Three rosters. (i) 2, 5, 8, 11, 14 and 17 each sent to 1. Verified: a function; domain those six numbers, range the single number 1 — the extreme case of repeated outputs. (ii) 2 with 1, 4 with 2, 6 with 3, 8 with 4, 10 with 5, 12 with 6, 14 with 7. Verified: a function; domain the even numbers 2 to 14, range 1 to 7. (iii) 1 with 3, 1 with 5, 2 with 5. Verified: not a function, since 1 opens twice.
• Exercise 2.3 Q3 (p. 38). The rule doubles its input and subtracts 5. Verified: at 0 it gives −5, at 7 it gives 9, at −3 it gives −11.
• Exercise 2.3 Q4 (p. 38). The Celsius-to-Fahrenheit rule, nine fifths of the input plus 32. Verified: at 0 it gives 32; at 28 it gives 252/5 + 32 = 82.4; at −10 it gives −18 + 32 = 14; and setting the output to 212 gives nine fifths of the input equal to 180, so the input is 100. This is the chapter's only example of a function with units attached, and the fourth part runs it backwards.

Figures to have open

• Fig 2.5 redrawn (p. 29), reused here so section 5 can show a relation failing only for want of one input. The element lists were read off the printed page.
• Fig 2.6 redrawn (p. 29) for section 6, with both arrows out of 9 visible.
• A two-box test panel that can be ticked or crossed independently, reused across sections 5 to 8. Standard schematic.
• A number line from 0 to 10 carrying the two stretches of Miscellaneous Exercise Q1, with the overlap point marked and the two candidate outputs written above it. Standard schematic; the chapter prints no figure for this item.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.4 "Functions", pp. 30–31.
• Definition 5, p. 30; Definition 6, p. 31.
• Examples 10, 11 and 12, p. 31, and the discussion of Examples 7, 8 and 9 at the head of p. 31.
• Exercise 2.3 items 1, 3 and 4, p. 38.
• Miscellaneous Exercise on Chapter 2, items 1, 10 and 11, pp. 40–41.
• Summary, p. 41, for the function entry and the naming of domain and codomain.