Understanding Mathematical Functions: Key Concepts and Definitions
RELATIONSHIP: A relationship exists between two sets if there is a link between them. These sets may be different or may occur within the same set.
ROLE: This is a relationship with two key features:
- Existence: For each element of the set of departure, there is another element in the set of arrival to which it is linked.
- Uniqueness: Each element of the set of departure is bound to a single element in the set of arrival.
SET OF DEPARTURE: The first set involved in the relationship.
SET OF ARRIVAL: The second set to which the relationship refers.
DOMAIN: The set of all possible values that the independent variable can take.
IMAGE: The set of all values that the dependent variable takes.
ROOTS: The values of the domain whose image is zero.
RANGE OF GROWTH: A subset of the domain for which higher values of the independent variable correspond to higher values of the dependent variable. If x > a, then f(x) > f(a).
RANGE OF DECREASE: A subset of the domain for which higher values of the independent variable correspond to lower values of the dependent variable. If x > a, then f(x) < f(a).
ABSOLUTE MAXIMUM: A function reaches its absolute maximum at a point ‘a’ in the domain if, for all x in the domain where x ≠ a, the image of x is less than the image of a. ∀x ≠ a: f(x) < f(a).
ABSOLUTE MINIMUM: A function reaches its absolute minimum at a point ‘a’ in the domain if, for all x in the domain where x ≠ a, the image of x is greater than the image of a. ∀x ≠ a: f(x) > f(a).
RELATIVE MAXIMUM: A function reaches a relative maximum at a point ‘a’ if there is an interval containing ‘a’ such that for all x in that interval where x ≠ a, the image of x is less than the image of a. ∀x ≠ a -> f(x) < f(a).
RELATIVE MINIMUM: A function reaches a relative minimum at a point ‘a’ if there is an interval containing ‘a’ such that for all x in that interval where x ≠ a, the image of x is greater than the image of a. x ≠ a -> f(x) > f(a).
C+: The subset of the domain whose images are positive numbers.
C–: The subset of the domain whose images are negative numbers.
EVEN FUNCTION: A function is even if, for all values of x belonging to the domain, f(x) = f(-x).
ODD FUNCTION: A function is odd if, for all values of x belonging to the domain, f(-x) = -f(x).
PERIODIC FUNCTION: A function is periodic if there exists a number ‘p’ such that f(x + p) = f(x), where ‘p’ is the period.
INJECTIVE FUNCTION (One-to-One): A function is injective if different values of the domain have different images. x1 ≠ x2 -> f(x1) ≠ f(x2). f(x1) = f(x2) -> x1 = x2.
SURJECTIVE FUNCTION (Onto): A function is surjective when all elements of the set of arrival are images of the elements of the domain. The codomain must be equal to the image of the function. y = f(x).
LINEAR FUNCTIONS: f(x) = mx + b
SLOPE: The variation of the dependent variable per unit of the independent variable. M = (y2 – y1) / (x2 – x1). y – y1 = m(x – x1).
PARALLEL LINES: They have the same slope.
PERPENDICULAR LINES: Slopes are opposite and reciprocal.
DISCRIMINANT:
- If greater than 0, there are 2 real roots.
- If equal to 0, there is 1 real root.
- If less than 0, there are no real roots.
PIECEWISE FUNCTION: A function defined by multiple sub-functions, each applying to a certain interval of the main function’s domain.
ABSOLUTE VALUE (or MODULE): The distance of a number from zero. The distance between two numbers ‘a’ and ‘b’ is |a – b|.
SYSTEM OF EQUATIONS: A solution to a system of equations is a point (x, y) that satisfies all the equations. The solution set of the system of equations is the set of all points (x, y) that are solutions to all equations.
QUADRATIC FUNCTION: A function of the form f(x) = ax2 + bx + c, and its graph is a curve called a parabola. Symmetric values on a quadratic function are the values of the domain that have the same image. The vertex of a parabola is the point on the graph whose x-coordinate is not symmetric. This coordinate is in the middle of any pair of symmetric values.
If ‘b’ is zero, the value of ‘c’ indicates the vertical displacement on the y-axis. If ‘b’ and ‘c’ are not zero, the parabola will undergo a displacement of the vertex and its branches. When the equation is given as a square of binomials, it is called the canonical form, which shows the coordinates of the vertex. The factored form is presented by a product between two binomials. This form shows the coordinates of the roots, and to move to the polynomial form, apply the distributive property.
Factored Form: a(x – x1)(x – x2)
Canonical Form: a(x – xv)2 + yv
Polynomial Form: ax2 + bx + c
Xv = -b/2a