Worksheets › Class 9 › Relations and Functions (Advanced) › Cheat sheet
Class 9 Relations and Functions (Advanced): cheat sheet
CBSE Mathematics. A relation from A to B is any set of ordered pairs from A × B. A function is a relation that gives each element of A exactly one image.
Key ideas
- Ordered pairs are equal only when both entries match: (a, b) = (c, d) means a = c and b = d.
- A × B is all pairs (a, b) with a in A and b in B; in general A × B is not B × A.
- Domain: the first entries used; range: the second entries used; codomain: the whole of B.
- A relation is a function when no element of A is missing and none has two images.
- [x] is the greatest integer not more than x, so [−2.3] = −3; |x| is never negative.
- y = f(x) + c moves the graph up by c (down if c < 0); y = f(x + k) moves it left by k (right if k < 0).
Remember
| Size of a Cartesian product | n(A × B) = n(A) × n(B) |
| Number of relations from A to B | 2 to the power n(A) × n(B) |
| Distributive rules | A × (B ∩ C) = (A × B) ∩ (A × C); A × (B ∪ C) = (A × B) ∪ (A × C) |
| Linear function from two values | a = (f(q) − f(p)) ÷ (q − p), then b = f(p) − ap |
| Domains to remember | 1/x: x ≠ 0; √x: x ≥ 0; x², |x|, x³: all real x |
Watch out
- Treating [−2.3] as −2: the greatest integer function always rounds down.
- Calling a relation a function when one element of A has no image.
- Shifting y = f(x + 2) to the right instead of to the left.
Worked example
Find the domain and range of each function: (a) y = 1/(x − 2) (b) y = 7 − |x| (c) y = √(x − 2).
Working: the reciprocal cannot divide by 0 and is never 0; |x| ≥ 0, so y is at most 7; a square root needs a non-negative number and is never negative
Answer: (a) domain: all real x except 2; range: all real y except 0; (b) domain: all real x; range: y ≤ 7; (c) domain: x ≥ 2; range: y ≥ 0
Practise Relations and Functions (Advanced)
Revise with this page, then practise: every Relations and Functions (Advanced) sheet in the app is new, with an answer key and working.
Coming soon for Android and iPhone.