Summary of Function: Inputs and Outputs

FUNCTION TOPICS: INPUTS AND OUTPUTS

Understanding the concept of a function as a special relationship between two sets.
Identification and interpretation of ordered pairs (input, output).
Differentiation between domain (set of possible inputs) and codomain (set of possible outputs).
The image of the function is the set of all outputs produced by the inputs in the domain.

Function definition: f: X → Y | for every x in X, there exists a unique y in Y such that f(x) = y
Function notation: f(x) represents the output of function f for an input x.
Example of linear function: f(x) = mx + b, where m is the slope and b is the y-intercept.
Example of quadratic function: f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Mathematical Function: Relationship between two sets that associates each element of the first set with exactly one element of the second set.
Domain: Set of all possible inputs (values of x) for which the function is defined.
Codomain: Set of values that the function output (values of f(x)) can take.
Image: Subset of the codomain that is actually reached by the function values.
Input (x): Value inserted into the function, which is transformed to produce an output.
Output (f(x)): Result obtained after applying the function to the input.
Mapping: Process of associating elements of the domain with elements of the codomain.
Dependency Relation: Indicates that the output value depends exclusively on the input value.

Functions as Machines: Visualizing functions as 'machines' where we input an 'x' (input), and it gives us an 'f(x)' (output) helps in understanding the processing of functions.
Unique Relation: In a function, each input corresponds to one and only one output, never multiple outputs.
Domain and Image: Understanding that not necessarily all values of the codomain are reached by the function is fundamental; the image is what is actually reached.

Function Notation: Learning to read and write expressions like f(x) = 2x + 3, understanding that 'f' is the function name, 'x' is the input variable, and '2x + 3' is the rule that determines the output.
Evaluating Functions: Knowing how to substitute values in place of 'x' and calculate the result is crucial to find function outputs, for example: for f(x) = x², if x = 3, then f(3) = 3² = 9.
Graph Analysis: Understanding function graphs allows visualizing how the input affects the output, facilitating the understanding of the function's behavior.

Linear Function: f(x) = 2x + 1. For input x = 5, the output is f(5) = 2(5) + 1 = 11.
- Shows a direct and constant relationship where the graph would be a straight line.
Quadratic Function: f(x) = x² - 4x + 4. For input x = 2, the output is f(2) = 2² - 4(2) + 4 = 0.
- Illustrates a relationship where outputs vary non-linearly in relation to inputs.
Interpreting Graphs: Given the graph of f(x) = x², identify that as x increases, the output f(x) grows squared.
- Reinforces the visual understanding of the relationship between input and output.

Functions: Relationships that map each input (x) to a single output (f(x)).
Domain: Set of all possible inputs for the function.
Codomain: Set of all outputs that the function can produce.
Image: Set of outputs that are effectively reached by the function.
Function Notation: The form f(x) represents the function output for the input x.
Function Behavior: The relationship between x and f(x) can be direct, inverse, linear, quadratic, etc.

Each input in a function has a unique corresponding output, with no duplicity.
Understanding the notation and the rule that defines the function is essential to calculate its output.
The ability to evaluate functions is built through the practice of substituting inputs and calculating outputs.
Analyzing graphs helps visualize and understand how inputs affect outputs.
Applying functions to practical problems develops logical reasoning and data interpretation skills.