Dominio y rango de una función

What are Domain and Range of a Function?

Introduction to Domain and Range

• The video introduces the concepts of domain and range in functions, aiming for a comprehensive understanding through three different explanations: using sets, functions, and graphs.

Understanding with Sets

• A simple set is presented as the output set (images), where the function f(x) = x^2 is used to illustrate how input values transform into output values.

• For example, when inputting 2 into the function, 2^2 = 4 . This establishes that the domain consists of all numbers from which outputs can be derived.

Exploring Input and Output Values

• The speaker explains that both positive and negative inputs yield the same output; for instance, both -1 and 1 produce an image of 1 when squared.

• The domain is defined as all possible input values (-3, -2, -1, 0, 1, 2, 3), while the range includes only those outputs that result from these inputs (0, 1, 4, and 9).

Conceptualizing Functions as Machines

• The function is likened to a machine where inputs (domain) are processed to produce outputs (range).

• An example with real numbers illustrates this concept: if you input number one into a square root function ( sqrtx ), it returns one.

Valid Inputs for Functions

• When testing other numbers like four or five in the square root function:

• Inputting four yields two.

• Inputting five results in approximately 2.23.

• However, negative numbers cannot be processed by this machine since they lead to errors due to undefined square roots.

Understanding Domain and Range Through Graphing

Visual Representation of Functions

• The speaker emphasizes that all functions can be graphed. In graphing:

• The domain corresponds to x-values where there is visible data on the graph.

• The range corresponds to y-values represented by points on the graph.

Identifying Domain and Range from Graphs

Understanding Domain and Range in Graphs

Introduction to the Graph

• The discussion begins with identifying where the drawing starts on the graph, specifically at the number -3. The speaker emphasizes that there is a drawing up to this point, indicating the limits of the graphical representation.

Determining Domain

• The domain of the function, referred to as f(x) , includes all numbers from -3 (inclusive) to 3 (exclusive). This concept ties back to interval notation, distinguishing between open and closed intervals.

Analyzing Range

• To find the range, one must examine the vertical axis (y-axis). The speaker notes that while there may not be any graph below a certain point, it starts appearing at -3 on the y-axis.

• The range begins at -3 (inclusive), as indicated by where the line crosses this value, and extends up to 4 (exclusive). This highlights how both domain and range are defined through specific endpoints.

Practice Exercise

• A practice exercise is introduced for viewers to determine both domain and range based on a provided graph. Viewers are encouraged to pause and analyze before checking their answers.

Summary of Findings

• For another example discussed, the domain ranges from -6 (open interval) on the left side to 9 (closed interval) on the right side.