Each standard function is pinned down by its picture as much as by its rule

What to assume they know

• The one-output rule that promotes a relation to a function — Definition 5, and what makes a relation a function
• What goes in, what comes out, and what was merely allowed to come out — domain, range and codomain
• Plotting points on axes from a table of values, and joining them into a curve
• Powers with whole and fractional exponents, and index notation
• Absolute value from earlier classes, at the level of stripping a sign
• Factorising a quadratic, for the domain examples that come later in the chapter

What they should be able to do

• Name the seven functions §2.4.1 introduces and say, for each, whether one expression covers it or it is stated by cases
• State the domain and range of each named function as the chapter gives them
• Complete a table of values from a rule and plot the resulting graph, as Examples 13 and 15 require
• Decide whether a stated expression defines a polynomial function, and identify the feature that disqualifies one
• Explain why the reciprocal function must drop one input, and what happens to its graph there
• Write the modulus, signum and greatest integer functions as case statements, and match each case to the part of the graph it draws
• Read from the greatest integer graph which end of each step belongs to it, and compute the function's value at a negative non-integer
• State what the Remark on p. 38 calls a linear function, and place the identity and constant functions inside that description

Where it usually goes wrong

• "A polynomial can carry any exponent." The chapter's own counter-example turns on a fractional one. Coefficients are unconstrained — one printed qualifying example uses root two — while exponents must be whole numbers from zero up.
• "The range of the squaring function is all the real numbers." It is the squares, so nothing below zero. The chapter writes it as a set of squares rather than as an interval, which is worth spelling out.
• "The modulus of a negative input is found by deleting the minus sign." The clause says the value is the negative of the input, and that comes out positive precisely because the input was negative. The deletion story stops working the moment the input is an expression rather than a number.
• "The signum function is undefined at zero." It is defined there and its value is zero. Only the alternative formula printed in Fig 2.14 has to exclude that input, because that formula divides by the input.
• "The greatest integer function rounds." It never rounds up. At 2.7 it gives 2, and at −1.5 it gives −2, not −1 — the two most productive errors, and both follow from taking the largest integer that does not exceed the input.
• "Fig 2.15 has no step between 0 and 1." It does; that step lies along the horizontal axis and is not drawn in its own colour. A student counting drawn segments will find six and conclude the function skips a stretch.
• "The reciprocal function has range all the real numbers, since it takes big and small values." No input produces zero, which is why the range drops zero as well as the domain, and why the two branches never meet the horizontal axis.
• "Identity and constant are unrelated to the linear function." Both fit the form in the Remark on p. 38 — one with no added number, the other with no multiple of the input. The chapter never joins them up.

Questions to check understanding

• Complete a table of values from a stated rule and draw the graph
• State the domain and range of each of the named functions
• Decide whether a given expression defines a polynomial function, with a reason
• Evaluate the greatest integer function at positive and negative non-integers
• Write the modulus or the signum function as a case statement, and identify the value at the boundary input
• Identify a named function from its graph alone
• Find the range of a simple rule on a restricted domain, as Exercise 2.3 Q5 asks
• Given a linear function through two known points, find the multiplier and the added number

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data. All figure descriptions were read off the printed pages, since the axis numbering and the curve labels sit inside the artwork.

• Identity, Fig 2.8 (p. 32). Domain and range both the real numbers. The drawing shows a straight line through the origin rising left to right; the axes are numbered in twos out to 8 and −8 on both, and the curve label sits beneath the frame.
• Constant, Fig 2.9 (p. 32). Domain the real numbers, range the one-element set holding the chosen constant. The drawing is instanced at the value 3: a horizontal line crossing the vertical axis between the 2 and 4 marks, with the same axis numbering as Fig 2.8.
• Polynomial (p. 33). The chapter prints two qualifying examples — the input cubed minus its square plus 2, and the input to the fourth plus root two times the input — and one that does not qualify, the input raised to two-thirds plus twice the input. It asks why and leaves the answer open. Verified: the disqualifier is the exponent two-thirds, since the definition admits only whole-number powers from zero upward. Note that the qualifying second example carries an irrational coefficient and still qualifies, because the definition constrains the exponents and lets the coefficients be any real numbers — a contrast worth drawing.
• Example 13 and Fig 2.10 (p. 33). The squaring function tabulated at −4, −3, −2, −1, 0, 1, 2, 3, 4. Verified: the outputs are 16, 9, 4, 1, 0, 1, 4, 9, 16. The chapter's domain is the real numbers, and it writes the range as the set of squares of real numbers rather than as an interval. The figure draws a narrow upward curve resting on the origin, on axes numbered as before.
• Example 14 and Fig 2.11 (p. 34). The cubing function, with these values printed: 0 goes to 0, 1 to 1, −1 to −1, 2 to 8, −2 to −8, 3 to 27, −3 to −27. Verified. The figure draws an S-shaped curve through the origin, and the values at ±3 lie far outside the frame's range of ±8, so the drawn curve leaves the top and bottom of the picture — worth saying, because students try to read the whole function off the visible part.
• Rational and Example 15 (pp. 34–35). The reciprocal function, declared on the real numbers with zero removed. The chapter's table gives nine inputs: −2, −1.5, −1, −0.5, 0.25, 0.5, 1, 1.5, 2. Verified: the outputs are −0.5, about −0.67, −1, −2, 4, 2, 1, about 0.67, 0.5. Two of them are rounded to two decimals in the printed table, since the exact values there are two thirds with a sign. The chapter states that the domain and the range are both the real numbers with zero removed. Fig 2.12 draws two separate branches, one in the upper right and one in the lower left, neither touching either axis.
• Modulus and Fig 2.13 (p. 35). Two clauses: the input itself when the input is at least zero, and the negative of the input when the input is below zero. Verified: the boundary input zero is assigned to the first clause, and both clauses would give zero there anyway, so the assignment does not change any value — unlike the piecewise item in the Miscellaneous Exercise, where it does. The graph is two rays meeting in a corner at the origin.
• Signum and Fig 2.14 (pp. 35–36). Three clauses: 1 above zero, 0 at zero, −1 below zero. The chapter states the domain as the real numbers and the range as the three-element set holding −1, 0 and 1. Read off the figure: the two horizontal rays are drawn with a small hollow circle at the end nearest the vertical axis, marking that neither ray reaches it, and the label line beneath the frame gives a second formula for the same function — the modulus of the input divided by the input, for every input but zero, with the value zero supplied separately at zero. That second formula appears only in the figure lettering and not in the running text.
• Greatest integer and Fig 2.15 (p. 36). The chapter prints four instances of the rule: the value is −1 across the stretch from −1 up to but not including 0; 0 from 0 up to but not including 1; 1 from 1 up to but not including 2; 2 from 2 up to but not including 3. Read off the printed page: the drawing carries six horizontal steps, at heights −3, −2, −1, 1, 2 and 3, each ending in a hollow circle at its right-hand end and each beginning without a marked dot. The step of height 0, which the printed rule covers, runs along the horizontal axis itself and is not drawn as a separate segment. Verified from the printed rule: the value at −1.5 is −2, because −1.5 falls in the stretch running from −2 up to but not including −1.
• Example 18 and Fig 2.16 (p. 38). The rule adds 10 to the input. Printed values: 0 goes to 10, 1 to 11, 2 to 12, 10 to 20, −1 to 9, −2 to 8, −10 to 0. Verified. The figure is a rising straight line with two points labelled inside the artwork, one on the vertical axis at height 10 and one on the horizontal axis at −10. The Remark directly beneath is where this chapter defines the linear function.
• Example 22 and Fig 2.17 (p. 39). A three-clause rule: one minus the input below zero, the value 1 at zero, and the input plus one above zero. Printed values: −4 goes to 5, −3 to 4, −2 to 3, −1 to 2, and 1 to 2, 2 to 3, 3 to 4, 4 to 5. Verified: both outer clauses tend to 1 as the input approaches zero and the middle clause supplies exactly 1, so the three pieces meet without a gap. The figure draws a V: two rays meeting at height 1 on the vertical axis, the left one falling as the input increases towards zero and the right one rising away from the vertex, each labelled with its own clause inside the artwork. Do not let anyone redraw it as a pair of parallel rising segments — the falling left arm is the whole point of the first clause.

Figures to have open

• Redrawn versions of Figs 2.8 through 2.15 (pp. 32–36) as schematics on consistent axes. These are the chapter's own figures and the topic cannot be taught without them; redraw rather than reproduce.
• Fig 2.14 needs its hollow circles drawn large enough to read, and needs the second formula from its label line carried across, since that formula is nowhere else in the chapter.
• Fig 2.15 needs a version in which the step lying on the horizontal axis is lifted or coloured so it can be seen, alongside a faithful version showing what the book actually prints.
• Fig 2.16 (p. 38) with its two labelled intercepts, for the linear-function section.
• A side-by-side sorting frame holding all seven functions at once, for sections 1 and 12. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.4.1 "Some functions and their graphs", pp. 32–36, items (i) to (vii).
• Examples 13, 14 and 15, pp. 33–35.
• Example 18 and the Remark defining the linear function, p. 38.
• Example 22, p. 39, for a three-clause rule whose pieces meet.
• Exercise 2.3 items 2 and 5, p. 38.
• Figs 2.8 and 2.9 (p. 32), 2.10 (p. 33), 2.11 (p. 34), 2.12 and 2.13 (p. 35), 2.14 and 2.15 (p. 36), 2.16 (p. 38), 2.17 (p. 39).