Summary of "FUNCIONES Desde 0: Qué son, Dominio, Rango y Codominio EVAU"

Overview

The video explains functions from the ground up: what a function is, input vs output, domain, range and codomain. It uses a “machine” analogy — you give a number (input) and the machine returns a number (output) according to a rule.

Definitions and basic concepts

Function: a rule that assigns to each element of a set (the domain) a single element of another set (the range/codomain). Notation: f(x) denotes the function value at input x.
Independent variable: usually x (the input).
Dependent variable: usually y or f(x) (the output, depends on x).
Domain: the set of all permissible inputs x for which the function is defined.
Range (image): the set of actual outputs produced by the function when x runs through the domain.
Codomain: the target set in which outputs lie as declared with the function; it can be larger than the actual range.

Example: for sine sin(x) the range (actual outputs) is [-1, 1], but the codomain could be declared as all real numbers if one chose.

Examples and specific points

Linear: f(x) = 2x + 1
- Domain: all real numbers.
- Example outputs: f(1) = 3, f(-2) = -3.
- Range: all real numbers.
Square root: f(x) = sqrt(x − 3)
- Domain: require radicand ≥ 0 → x − 3 ≥ 0 → domain [3, ∞). (Square root of 0 is allowed.)
Rational function with denominator x^2 − 4:
- Denominator zero at x = 2 and x = −2 → domain is all real numbers except ±2.
Root in denominator: e.g., 1 / sqrt(x^2 − 4)
- Need radicand > 0 (radicand = 0 would make the denominator zero) → domain (−∞, −2) ∪ (2, ∞) (endpoints excluded).
Range examples:
- 2x + 1: range = all real numbers.
- sin(x) and cos(x): range = [-1, 1].
- Squaring function f(x) = x^2: domain all reals, range [0, ∞).

Domain rules by function type

Polynomials, odd-root radicals, exponentials, sin, cos, arctan: domain = all real numbers.
Rational functions: domain = all real numbers except values that make the denominator 0.
Even-root radicals (square root, 4th root, …): exclude x values that make the radicand negative (require radicand ≥ 0).
Logarithmic functions: require the argument to be positive (argument > 0).
Tangent: undefined where cosine = 0 → exclude x = π/2 + kπ (k ∈ Z).
Arcsine and arccosine: domain = [-1, 1].

Practical methodology — how to determine the domain

For a given algebraic function f(x):

Identify operations that impose restrictions:
- Division: find values that make a denominator 0 → exclude them.
- Even-indexed root (square root, 4th root, …): require the radicand ≥ 0.
- Logarithm: require the argument > 0.
- Trigonometric expressions: check where denominators or trig identities cause undefined values (e.g., tan x undefined where cos x = 0).
Solve the inequalities/equations from those restrictions to find forbidden or allowed x-values.
Combine all restrictions (take the intersection of allowed sets) to obtain the final domain.
Check endpoints:
- If an endpoint yields division by zero, exclude it.
- If an endpoint yields a zero inside an even root, include it (since root(0) = 0).

How to find the range (basic approaches)

Analyze the algebraic form and behavior of f(x):
- Linear functions produce all real outputs.
- Quadratics opening upwards have a minimum → range [min, ∞).
- Trigonometric functions have known bounded intervals.
Alternatively, set y = f(x) and try to solve for x; deduce which y values are possible for real x.

Clarifications and corrections

Logarithm domain: the video’s phrasing “not negative” should be interpreted strictly as “argument must be positive” (logarithm is undefined at 0).
Tangent singularities: occur at x = π/2 + kπ (k ∈ Z) because cos x = 0 there.
Codomain vs range: the codomain is the set you declare the function maps into (it may be larger); the range is the actual set of outputs.

Closing / next steps

The video is an introductory class. Future videos will cover characteristics of functions in depth: symmetry, periodicity, compositions, inverse functions, etc.
The presenter has a Facebook community for exam preparation (EVAU) and problem help — link in the video description.