• In a relation, two things are related to each other by a relating phrase.
• Mathematically, a relation is a set of ordered pairs. If A and B are two non-empty sets, then the relation from A to B is a subset of A × B that satisfies the relating phrase.
• If A and B are any sets and R ⊆ A × B, we call R a binary relation from A to B or a binary relation between A and B.A relation R ⊆ A × A is called a relation in A or a reaction on A.
• The set {x: (x, y) ∈ R for some y} is called the domain of the relation R.
  The set {y: (x, y) ∈ R for some x} is called the range of the relation R.
• A function is a special type of a relation in which each x-coordinate is paired with exactly one unique y-coordinate.
• A function from A to B can sometimes be denoted as f: A→B, where the domain of f is A and the range of f is a subset of B, in which case B contains the images of the elements of A by the function f.
• Let f and g be functions. We define the sum f + g, the difference f – g, the product fg, and the quotient

as:

• If a and b are fixed real numbers, a≠ 0, then f (x) = ax + b for x ∈ ℝ is called a linear function. If a = 0 then f (x) = b is called a constant function. Sometimes linear functions are written as y = ax + b.
• In f (x) = ax + b for a ≠ 0, x ∈ ℝ, a represents the slope, (0, b) represents the

• Two lines with equal slopes are parallel. Distinct parallel lines never intersect.
• Two lines with different slopes intersects at one and only one point.
• The distance d between points (x₁, y₁) and (x₂, y₂) is given by

• A function defined by f(x) = ax²+ bx + c, (a, b, c∈ℝ and a≠ 0) is called quadratic function. a is called the leading coefficient.
• We can sketch the graph of a linear function by using either a table of values, or the x- and y-intercepts.
• We can sketch the graph of a quadratic function by using either a table of values or the shifting rules.
• A quadratic function f(x) = ax² + bx + c can be written as

Moreover, the graph of this quadratic function is a parabola with vertex (h,k), line (axis) of symmetry x=h .

• If a > 0, the parabola opens upward. In this case rang(f ) = {y∈ℝ | y ≥ k }; i.e., the minimum value of f (x) is k which occurs at x=h.
• If a < 0, the parabola opens downward In this case rang(f ) = {y∈ℝ | y ≤ k }; i.e., the maximum value of f (x) is k which occurs at x=h.