Formula Sheet: Relations and Functions
This chapter explores the concepts of relations and functions in mathematics, focusing on how to connect pairs of objects from different sets and the significance of functions in describing these relationships.
Relations and Functions – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Relations and Functions chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formula sheet
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
P × Q = {(p, q) : p ∈ P, q ∈ Q}
P × Q represents the Cartesian product of sets P and Q, where each element of P is paired with each element of Q. This is fundamental in defining relations.
Domain(R) = {x : (x, y) ∈ R}
The domain of relation R consists of all first elements from the ordered pairs. It represents all possible inputs for the relation.
Range(R) = {y : (x, y) ∈ R}
The range of relation R consists of all second elements from the ordered pairs. This indicates all possible outputs produced by the relation.
n(A × B) = n(A) × n(B)
If set A has p elements and set B has q elements, then the Cartesian product A × B will contain pq elements. This is crucial for counting relations.
f: A → B
A function f from set A to set B signifies that each element in A corresponds to precisely one element in B. It's a specific type of relation.
f(a) = b
This indicates that for function f, the input a from the domain A yields output b in the codomain B. It emphasizes the concept of image in function theory.
R ⊆ A × B
A relation R from set A to set B is a subset of the Cartesian product A × B, describing a relationship between elements of A and B.
x ∈ A and y ∈ B
We denote that elements x belong to set A and elements y belong to set B. This foundation is crucial for understanding relations and functions.
f + g: X → R, (f + g)(x) = f(x) + g(x)
This defines the pointwise addition of two functions f and g over set X. It's used extensively in operations on functions.
f - g: X → R, (f - g)(x) = f(x) - g(x)
This defines the pointwise subtraction of the function g from function f. It follows similar principles as addition.
Equations
y = mx + c
This equation represents a linear function where m is the slope and c is the y-intercept. It is foundational in algebra for graphing straight lines.
f(x) = a0 + a1x + a2x² + ... + anxⁿ
This represents a polynomial function of degree n. Each coefficient a_i corresponds to the x raised to the power of i.
f(x) = kx, k ∈ R
This defines a constant function where k is a constant multiplier. It's essential in understanding transformations of functions.
f(x) = 2x + 1
This linear function indicates that for every x, the output is double x plus one. It's a classic example used in function applications.
g(x) = x²
This quadratic function demonstrates the parabolic relationship where output is the square of the input. It's fundamental in algebra.
h(x) = 1/x, x ≠ 0
This defines a rational function. It's essential in higher algebra and calculus to work with domains exceeding standard real numbers.
R = { (x, y) : y = f(x) }
A way to represent a relation in terms of a function f. It indicates that y is determined by the input x through function f.
f(x) = |x|
The modulus function reflects all negative inputs to their positive counterparts, crucial in understanding numeric boundaries.
f/g, g(x) ≠ 0
This represents the division of two functions where g(x) is non-zero. Important in the analysis of rational functions and their limits.
f(x) = 2x - 3
This linear equation delineates the relationship between x and its corresponding outputs through a slope and intercept.
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