Combining two functions point by point, and the one case that fails

What to assume they know

• The one-output rule that promotes a relation to a function — Definition 5, and Definition 6's real function
• Each standard function is pinned down by its picture as much as by its rule — the named functions, and how a domain gets restricted
• Expanding a product of a monomial with a linear expression, and collecting terms
• Index laws for fractional and negative exponents, including the square root written as a one-half power
• Solving a linear equation to find where an expression takes the value zero
• Factorising a quadratic, for finding where a denominator vanishes

What they should be able to do

• State how each of the five constructions in §2.4.2 defines the value of the new function at a given input
• Explain why four of the five require two functions defined on one common set, and why the scalar multiple is the exception, taking one function and a number
• Form the sum, difference, product and quotient of two given real functions and simplify each result
• Identify the inputs a quotient must exclude by solving for the zeros of the lower function, and state the excluded set explicitly
• Explain why a scalar multiple scales the output and not the input
• Use index laws to simplify a product or quotient involving a square root, and show that the exclusion survives the simplification
• Distinguish the pointwise product of two functions from any operation that feeds one function into the other

Where it usually goes wrong

• "The quotient is undefined when the lower function is the zero function." The condition is checked one input at a time. In Example 16 the lower rule is nowhere near constant and only a single input has to go.
• "The excluded input comes from the upper function." It comes from the zeros of the lower one. In Example 16 the upper rule is zero at the input zero, and zero is not excluded.
• "Once you simplify, the exclusion goes away." Example 17's quotient tidies into a negative half power, which is still undefined at zero. The excluded input belongs to the function, not to the way it happens to be written.
• "The product of two functions means feeding one into the other." It means multiplying their values at the same input. This chapter defines no operation that feeds one function into another.
• "A scalar multiplies the input." It multiplies the output. The identity function will not expose the difference, and neither will any rule that just multiplies its input by a fixed number: scaling before and scaling after both come to the same constant times the same input. Put the constant function of Each standard function is pinned down by its picture as much as by its rule §3 beside it instead — scaling its output changes the value and scaling its input does nothing at all.
• "The two functions need the same range." They need the same set of inputs. Example 17's two rules produce different outputs at almost every input and combine without difficulty.
• "Four operations are safe and division is risky, so division is a special kind of construction." It is defined by the same manoeuvre as the other four. What differs is a fact about the real numbers, not a fact about functions.

Questions to check understanding

• Given two real functions, write down their sum, difference, product and quotient, simplifying each
• State the input or inputs a stated quotient must exclude, showing the equation solved to find them
• Find the domain of a function given as one polynomial over another, by factorising the lower one
• Simplify a quotient involving a square root using index laws, and say why the exclusion is still needed afterwards
• Explain in words why four of the five constructions need no condition and the fifth does
• Evaluate a combined function at a stated input by evaluating both parts first

Examples worth working on the board

Values marked verified are an added algebra on the chapter's stated rules. §2.4.2 prints no figure of its own; the drawings this topic wants are all schematics.

• The five definitions (§2.4.2, pp. 36–37). Both functions run from a common set of real inputs to the real numbers. The sum's value at an input is the sum of the two values; the difference's is their difference; the scalar multiple's is the number times the one value; the product's is the product of the two values; the quotient's is the first value over the second, on the condition that the second is not zero.
• Example 16 (p. 37). The first rule squares its input; the second doubles it and adds one. Verified: sum — the square plus twice the input plus one; difference — the square minus twice the input minus one; product — expanding the square against the linear rule gives twice the input cubed plus the square; quotient — the square over the linear rule, with one input excluded. Verified separately: the linear rule is zero when twice the input equals −1, so the excluded input is −1/2, which is what the chapter prints beside the quotient. Worth showing and not in the chapter: the sum factorises as the square of one more than the input, while the difference does not factorise over the integers. That contrast makes the point that the constructions are defined by their values, not by any tidiness in the resulting expression.
• Example 17 (p. 37). The first rule takes the square root of its input; the second returns the input unchanged. Both are declared on the non-negative real numbers. Verified: sum — root of the input plus the input; difference — root of the input minus the input; product — the root times the input, which by the index laws is the input raised to three halves; quotient — the root over the input, which is the input raised to minus one half, with zero excluded. Verified separately: the lower rule vanishes only at zero, so zero is the only input removed; and the simplified form, a negative half power, is undefined at zero as well, so the exclusion survives the simplification rather than being hidden by it. This is the example where two restrictions overlap — the declared non-negative domain and the excluded zero.
• Miscellaneous Exercise Q7 (p. 40). Two rules on the real numbers: one adds 1 to the input, the other doubles it and subtracts 3. Verified: the sum is three times the input minus 2; the difference is 4 minus the input; the quotient is the first over the second, excluding the input where twice the input equals 3, that is 3/2.
• Example 21 (p. 39) as a preview of the same skill. The domain of a quotient of two quadratics is asked for, and the chapter factorises the lower one into two linear factors to locate its zeros. Verified: the lower expression is zero at the two inputs 1 and 4, and the domain is the real numbers with those two removed. The move is identical to the one Example 16 makes with a linear denominator; only the number of excluded inputs changes.
• Miscellaneous Exercise Q3 (p. 40) for the same move at a third difficulty. The lower expression is the square of the input, minus eight times the input, plus 12. Verified: it factorises into the input minus 2 and the input minus 6, so the two excluded inputs are 2 and 6.
• The Summary restatement (p. 42). All five constructions reappear, with the quotient carrying the same condition on the lower function. Verified against the two pages: the scalar is written with a different letter in the Summary from the one §2.4.2 uses, so a student comparing the two pages will meet the same rule under two names. Say so rather than letting them notice it alone.

Figures to have open

• A twin-track schematic: one input feeding two function boxes, their two outputs arriving at a combining node. Reusable for all five constructions and the backbone of sections 3 to 6.
• A number line able to carry a hollow circle at an excluded input, used for Example 16, Example 17 and the two quadratic-denominator items. Standard schematic.
• A two-column comparison panel for §2.4.2 against the Summary. Standard schematic.
• The chapter prints no figure in §2.4.2 or in the Summary's algebra entry, so nothing here is redrawn from the book.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.4.2 "Algebra of real functions", pp. 36–37, items (i) to (v).
• Examples 16 and 17, p. 37.
• Example 21, p. 39, for a quotient's domain found by factorising.
• Miscellaneous Exercise on Chapter 2, items 3 and 7, p. 40.
• Summary, p. 42, for the restatement of all five constructions.
• Definition 6, p. 31, for the meaning of a real function.