Relations and Functions — Class 11 ISC Mathematics revision notes
A 4-page visual revision sheet for Relations and Functions, written for Class 11 ISC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Relations and Functions in one look
ordered pairs, and the rule that a function may not break
From a product to a relation
- A x B is the set of ordered pairs (a, b) with a in A and b in B, so n(A x B) = n(A) x n(B)
- ordered means the position carries meaning: (1, 2) is not (2, 1), and two pairs are equal only when both coordinates match
- a relation R from A to B is any subset of A x B, which is the entire definition; a set of m by n pairs therefore has 2^mn possible relations
- domain is the set of first coordinates that actually appear, range the set of second coordinates, and codomain is all of B whether used or not
A function, and the one rule
- a function from A to B is a relation in which EVERY element of A appears as a first coordinate exactly once
- so two conditions: nothing in the domain is left out, and nothing in the domain has two images
- the vertical line test is that rule drawn: a vertical line may cut the graph at most once
- two elements of A may share the same image, which is allowed and is what makes a function many to one rather than illegal
- the range is a SUBSET of the codomain, and they are equal exactly when the function is onto
Finding a domain, which is most of the marks
- the domain is every real input for which the formula gives a real answer, so three things are being excluded
- a denominator may not be zero, so solve denominator = 0 and remove those points
- an even root needs a non negative inside, so solve the inequality rather than an equation
- a logarithm needs a strictly positive argument, and the base must be positive and not 1
- when several conditions apply, the domain is their INTERSECTION, and writing it as an interval is what the answer must look like
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Relations and Functions sheet covers
- From a product to a relation
- A function, and the one rule
- domain codomain and range
- codomain
- Finding a domain, which is most of the marks
- The standard functions, with domain and range
- The step and modulus functions, drawn carefully
- Algebra of functions
- The three kinds of relation on one set
- Testing a relation without hand waving
- The empty and universal relations
- Graphs worth being able to sketch from memory
- Even, odd and periodic
- Range, found honestly
- Counting relations and functions
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Relations and Functions the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.