Relations and Functions ISC Class 11: Notes and Graphs

What Relations and Functions is about

Weightage: Relations and Functions belongs to the unit Sets and Functions, which CISCE weights at 18 of the 80 marks in the ISC Class XI Mathematics Paper I (Theory). Class XI is examined by the school, so your school sets the paper within this weightage.Source: CISCE ISC Mathematics syllabus, examination year 2028, Class XI unit weightage

Key concepts

Ordered pairs and Cartesian products

An ordered pair (a, b) has a definite order: (1, 2) ≠ (2, 1). Two ordered pairs are equal only if their first elements are equal and their second elements are equal. The Cartesian product of non-empty sets A and B is A × B = {(a, b) : a ∈ A, b ∈ B}. If either set is empty, A × B = ∅. In general A × B ≠ B × A, but n(A × B) = n(B × A) = n(A) × n(B). R × R = {(x, y) : x, y ∈ R} represents the coordinate plane; R × R × R = {(x, y, z)} represents three-dimensional space (ordered triples).

Worked example

If A = {1, 2} and B = {a, b, c}, write A × B and find n(A × B) and n(B × B).

Try it first, then show the solutionSolution

1. A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
2. n(A × B) = 2 × 3 = 6; n(B × B) = 3 × 3 = 9

Answer: 6 ordered pairs; 9

Relations: domain, co-domain and range

Any subset R of A × B is called a relation from A to B; we write (a, b) ∈ R or a R b. It may be given in roster form, in set-builder form, or by an arrow diagram (two ovals with arrows from each a to its related b). The domain of R is {a : (a, b) ∈ R}; the co-domain is the whole set B; the range is {b : (a, b) ∈ R}, a subset of the co-domain. If n(A) = p and n(B) = q, the number of subsets of A × B, and hence of relations from A to B, is $2^{pq}$. A relation on A is a subset of A × A.

Worked example

A relation R on A = {1, 2, 3, 4, 5, 6} is defined by R = {(a, b) : b = a + 2}. Write R in roster form and find its domain and range.

Try it first, then show the solutionSolution

1. Pairs with b in A: (1, 3), (2, 4), (3, 5), (4, 6)
2. Domain = {1, 2, 3, 4}; range = {3, 4, 5, 6}; co-domain = A

Answer: R = {(1, 3), (2, 4), (3, 5), (4, 6)}; domain {1, 2, 3, 4}; range {3, 4, 5, 6}

Functions as special relations

A relation f from A to B is a function (mapping) if every element of A is related to exactly one element of B. We write f : A → B and y = f(x); y is the image of x and x a pre-image of y. A is the domain, B the co-domain, and the set of images f(A) the range (⊂ B). Two failures make a relation not a function: an element of A with no image, or an element with two or more images. Several elements may share an image. Vertical line test: a graph in the plane is the graph of a function of x if no vertical line meets it more than once.

Worked example

Which of these relations from A = {1, 2, 3} to B = {4, 5, 6} are functions? R₁ = {(1, 4), (2, 5), (3, 5)}; R₂ = {(1, 4), (1, 5), (2, 6), (3, 6)}; R₃ = {(1, 4), (2, 6)}.

Try it first, then show the solutionSolution

1. R₁: each of 1, 2, 3 has exactly one image: a function (2 and 3 share 5, which is allowed).
2. R₂: 1 has two images: not a function.
3. R₃: 3 has no image: not a function from A to B.

Answer: Only R₁ is a function

Domain and range of real functions

For a real function given by a formula, the domain is the largest set of real x for which f(x) is a real number. Rules: a denominator cannot be zero; an even root needs a non-negative radicand; a logarithm needs a positive argument. To find the range, set y = f(x), solve for x in terms of y, and find the values of y for which x is real and lies in the domain; or use the graph. Examples: f(x) = 1/(x − 2) has domain R − {2} and range R − {0}; f(x) = √(9 − x²) has domain [−3, 3] and range [0, 3].

Worked example

Find the domain and range of f(x) = √(9 − x²).

Try it first, then show the solutionSolution

1. Need 9 − x² ≥ 0, i.e. −3 ≤ x ≤ 3: domain [−3, 3]
2. 9 − x² takes values from 0 (at x = ±3) to 9 (at x = 0), so √ takes values from 0 to 3
3. Range = [0, 3]

Answer: Domain [−3, 3]; range [0, 3]

Standard real functions and their graphs (1)

Constant function f(x) = c: domain R, range {c}; a horizontal line. Identity function f(x) = x: domain and range R; the line y = x. Polynomial function f(x) = a₀ + a₁x + … + aₙxⁿ: domain R (e.g. y = x² is a parabola with range [0, ∞)). Rational function f(x) = p(x)/q(x), q(x) ≠ 0: e.g. f(x) = 1/x has domain and range R − {0}, a rectangular hyperbola in the first and third quadrants. Modulus function f(x) = |x| = x for x ≥ 0 and −x for x < 0: domain R, range [0, ∞); a V-shape with its vertex at the origin. Signum function sgn x = |x|/x for x ≠ 0 and 0 for x = 0: values 1 (x > 0), 0 (x = 0) and −1 (x < 0); range {−1, 0, 1}.

Worked example

Evaluate |−4|, sgn(−5), sgn(0) and sgn(3.2). Write the range of f(x) = |x| + 2.

Try it first, then show the solutionSolution

1. |−4| = 4; sgn(−5) = −1; sgn(0) = 0; sgn(3.2) = 1
2. |x| ≥ 0, so |x| + 2 ≥ 2: range [2, ∞)

Answer: 4, −1, 0, 1; [2, ∞)

Standard real functions and their graphs (2)

Exponential function f(x) = $a^x$ (a > 0, a ≠ 1): domain R, range (0, ∞); its graph passes through (0, 1), rises for a > 1 (e.g. $e^x$, $2^x$) and falls for 0 < a < 1; the x-axis is an asymptote. Logarithmic function f(x) = ${\log }_{a}x$ (a > 0, a ≠ 1): domain (0, ∞), range R; it is the inverse of $a^x$, its graph passes through (1, 0) and is the mirror image of $a^x$ in the line y = x; the y-axis is an asymptote. Greatest integer function f(x) = [x], the greatest integer less than or equal to x: domain R, range Z; [2.7] = 2, [−2.3] = −3, [5] = 5. Its graph is a staircase of horizontal segments, closed (filled dot) at the left end and open at the right.

Worked example

Find [3.9], [−0.5], [−4] and the value of ${\log }_{2}8+{\log }_{10}0.01$.

Try it first, then show the solutionSolution

1. [3.9] = 3; [−0.5] = −1; [−4] = −4
2. log₂ 8 = 3; log₁₀ 0.01 = −2; sum = 1

Answer: 3, −1, −4; 1

Algebra of real functions

Let f : X → R and g : X → R with a common domain X ⊂ R. Then: sum (f + g)(x) = f(x) + g(x); difference (f − g)(x) = f(x) − g(x); product (fg)(x) = f(x)g(x); multiplication by a scalar (αf)(x) = αf(x); quotient $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$, defined only where g(x) ≠ 0. If f and g have different domains, the new function's domain is the intersection of the two domains (minus the zeros of g for a quotient).

Worked example

If f(x) = √x and g(x) = x² − 1, find (f + g)(4), (fg)(1) and the domain of f/g.

Try it first, then show the solutionSolution

1. (f + g)(4) = 2 + 15 = 17; (fg)(1) = 1 × 0 = 0
2. Domain of f is [0, ∞), of g is R; g(x) = 0 at x = ±1
3. Domain of f/g = [0, ∞) − {1}

Answer: 17; 0; [0, 1) ∪ (1, ∞)

Maths in real life: Relations and Functions

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Cartesian Products, Relations, Domain and Range
• Functions: Domain, Range, Graphs and Algebra

Relations and Functions formula sheet

Equality of ordered pairs

$(a,b)=(c,d)⟺a=c\text{and}b=d$

Order matters.

Size of a Cartesian product

$n(A\times B)=n(A)\times n(B)$

A × B ≠ B × A in general.

Number of relations

$2^{n(A)n(B)}$

Every subset of A × B is a relation.

Modulus and signum

$|x|=\{\begin{array}{ll}x,& x\ge 0\\ -x,& x<0\end{array},\text{sgn}x=\{\begin{array}{ll}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$

Ranges [0, ∞) and {−1, 0, 1}.

Greatest integer function

$[x]=n\text{if}n\le x<n+1,n\in Z$

[−2.3] = −3.

Exponential and logarithm

$y=a^x⟺x={\log }_{a}y;D(a^x)=R,R(a^x)=(0,\infty )$

Log: domain (0, ∞), range R.

Algebra of functions

$(f\pm g)(x)=f(x)\pm g(x),(fg)(x)=f(x)g(x),\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)},g(x)\ne 0$

Domain = intersection of domains.

Relations and Functions: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the school exam asks Relations and Functions

The question patterns to expect, and the method that works for each.

Tricks to remember Relations and Functions

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Every one, exactly one

  Function = every input gets exactly one output (no orphans, no twins).

  Use it when: Testing relations.

• Three domain rules

  No zero downstairs, no negative under an even root, only positive inside a log.

  Use it when: Finding domains quickly.

• GIF goes down

  [x] always rounds down to the integer on the left: [−2.3] = −3, not −2.

  Use it when: Greatest integer values.

• Swap for inverses

  aˣ and logₐx swap domain and range and reflect in y = x.

  Use it when: Exponential and log graphs.

• pq and 2 to the pq

  Pairs: pq. Relations: 2^(pq).

  Use it when: Counting questions.

Relations and Functions: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section D style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 5-mark question split into 1 + 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Relations and Functions

The slips examiners see most often, and what to write instead.

• ✗Mistake: Giving the co-domain as the range

  Students write the whole set B as the range.

  Wrong working

  R = {(1, 3), (2, 4)} from A to B = {3, 4, 5}: range = {3, 4, 5}.

  Right working

  Range = {3, 4} (images actually used); co-domain = {3, 4, 5}.

  ✓Do this: Range = set of second elements that appear.

  Usually costs: 1 mark.

• ✗Mistake: Wrong value of [x] for negative x

  Students drop the decimal part as if truncating.

  Wrong working

  [−2.3] = −2.

  Right working

  [−2.3] = −3 (greatest integer ≤ −2.3).

  ✓Do this: Picture the number line: go left to the nearest integer.

  Usually costs: 1 mark.

• ✗Mistake: Forgetting to exclude zeros of the denominator after simplifying

  Students cancel a factor and lose the excluded point.

  Wrong working

  f(x) = (x² − 9)/(x − 3) = x + 3, so domain and range are R.

  Right working

  Domain R − {3}; range R − {6}.

  ✓Do this: Find the domain from the original expression before simplifying.

  Usually costs: 1-2 marks.

• ✗Mistake: Calling a relation with an unused domain element a function

  Students check only that no element has two images.

  Wrong working

  {(1, 4), (2, 6)} from {1, 2, 3} to B is a function.

  Right working

  Not a function: 3 has no image.

  ✓Do this: Check both conditions: every element has an image, and only one.

  Usually costs: 1 mark.

• ✗Mistake: Using ≥ 0 instead of > 0 under a root in a denominator

  The root may be zero, but a denominator may not.

  Wrong working

  1/√(x² − 5x + 6): domain (−∞, 2] ∪ [3, ∞).

  Right working

  (−∞, 2) ∪ (3, ∞): end points make the denominator zero.

  ✓Do this: Combine the rules: root in a denominator needs > 0.

  Usually costs: 1 mark.

Practice questions on Relations and Functions

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   If n(A) = 4 and n(B) = 3, then n(A × B) is:
   (a) 7
   (b) 12
   (c) 64
   (d) 81

   Hint

   Multiply.

   Answer

   (b)

2. Q2 · 1 mark · MCQExam level

   The range of the signum function is:
   (a) R
   (b) {−1, 1}
   (c) {−1, 0, 1}
   (d) [−1, 1]

   Hint

   Value at 0.

   Answer

   (c)

3. Q3 · 1 mark · Very short answerExam level

   Find the value of [−3.7] + [3.7].

   Hint

   −4 and 3.

   Answer

   −4 + 3 = −1

4. Q4 · 1 mark · Very short answerExam level

   State the domain and range of f(x) = log₁₀ x.

   Hint

   Inverse of 10ˣ.

   Answer

   Domain (0, ∞); range R.

5. Q5 · 2 marks · Short answerExam level

   If A = {1, 2, 3} and B = {4}, write A × B and B × A. Are they equal?

   Hint

   Order matters.

   Answer

   A × B = {(1, 4), (2, 4), (3, 4)}; B × A = {(4, 1), (4, 2), (4, 3)}; not equal.

6. Q6 · 2 marks · Short answerHarder exam level

   Find the domain and range of f(x) = |x − 3|.

   Hint

   V-shape with vertex at x = 3.

   Answer

   Domain R; range [0, ∞).

7. Q7 · 2 marks · Short answerHarder exam level

   Find the domain of f(x) = 1/(x² − 4).

   Hint

   Denominator zero at ±2.

   Answer

   R − {−2, 2}.

8. Q8 · 2 marks · Short answerHarder exam level

   If f(x) = x + 1 and g(x) = x − 1, find (fg)(x), (f/g)(x) and the domain of f/g.

   Hint

   Exclude g(x) = 0.

   Answer

   (fg)(x) = x² − 1; (f/g)(x) = (x + 1)/(x − 1); domain R − {1}.

9. Q9 · 2 marks · Short answerHarder exam level

   A relation R on A = {1, 2, 3, 4, 5, 6} is defined by R = {(x, y) : y is divisible by x}. Write R in roster form and find its range.

   Hint

   Every x divides itself.

   Answer

   R = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(5,5),(6,6)}; range A.

10. Q10 · 3 marks · Short answerHarder exam level

    Find the range of f(x) = x/(x + 1), x ≠ −1.

    Hint

    Solve y = x/(x + 1) for x.

    Answer

    y(x + 1) = x gives x = y/(1 − y), real for y ≠ 1. Range R − {1}.

11. Q11 · 3 marks · Short answerHarder exam level

    Sketch the graph of f(x) = [x] for −2 ≤ x < 3 and state the values f(−1.5), f(0) and f(2.99).

    Hint

    Steps closed on the left.

    Answer

    Steps at heights −2, −1, 0, 1, 2 on [−2, −1), [−1, 0), [0, 1), [1, 2), [2, 3); f(−1.5) = −2, f(0) = 0, f(2.99) = 2.

12. Q12 · 5 marks · Long answerHarder exam level

    (a) Define a function as a special kind of relation. [1]
    (b) Find the domain and range of f(x) = √(16 − x²). [2]
    (c) Draw the graphs of y = 2ˣ and y = log₂x on the same axes and state their domains and ranges. [2]

    Hint

    Mirror images in y = x.

    Answer

    (a) A relation from A to B in which every element of A has exactly one image in B. (b) Domain [−4, 4]; range [0, 4]. (c) 2ˣ: through (0, 1), domain R, range (0, ∞); log₂x: through (1, 0), domain (0, ∞), range R; reflections in y = x.

13. Q13 · 3 marks · Short answerHOTS / competency

    Find all functions from A = {1, 2} to B = {a, b} and the number of functions from a set of m elements to a set of n elements.

    Hint

    Each element chooses its image.

    Answer

    {(1,a),(2,a)}, {(1,a),(2,b)}, {(1,b),(2,a)}, {(1,b),(2,b)}: 4 functions. In general nᵐ functions.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.n(A × B)?Show answer

   n(A) × n(B).

2. 2.What is a relation from A to B?Show answer

   Any subset of A × B.

3. 3.When is a relation a function?Show answer

   When every element of the domain has exactly one image.

4. 4.[−0.5]?Show answer

   −1.

5. 5.Domain of 1/x?Show answer

   R − {0}.

6. 6.Range of |x|?Show answer

   [0, ∞).

7. 7.Range of aˣ?Show answer

   (0, ∞).

8. 8.Domain of f/g?Show answer

   Common domain minus the zeros of g.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you write A × B and count its elements?
• ☐Can you find the domain and range of a relation?
• ☐Can you decide whether a relation is a function?
• ☐Can you find the domain of root, log and rational functions?
• ☐Can you find the range by solving y = f(x) for x?
• ☐Can you sketch |x|, sgn x and [x]?
• ☐Can you sketch aˣ and logₐx?
• ☐Can you find f + g, fg and f/g and their domains?

Relations and Functions: frequently asked questions