DOMÍNIO, CONTRADOMÍNIO e IMAGEM DA FUNÇÃO \Prof. Gis/ - AULA 8

Summary

In this lesson, the instructor explains key concepts of mathematical functions, focusing on the domain, codomain, and image (range). Through clear examples, the script demonstrates how to identify the domain (set of all inputs), codomain (set of possible outputs), and image (set of actual outputs) of a function. Using both a diagram and algebraic manipulation, students are guided through exercises to better understand the relationship between sets and how to determine these components. The video provides practical insights for mastering functions and offers valuable tips for solving related exercises.

Takeaways

• 😀 The domain of a function consists of the elements of the set A (the input values).
• 😀 The codomain of a function is represented by the set B (the possible output values).
• 😀 The image of a function consists of the elements of set B that are actually mapped to from set A.
• 😀 To determine the domain, codomain, and image, it's essential to know the function that defines the relationship between the two sets.
• 😀 A simple example was used where the function f(x) = x + 1 was applied to a set A with elements {1, 2, 3, 4, 5}.
• 😀 The relationship between set A and set B was demonstrated using a diagram of arrows to show how elements from A correspond to elements in B.
• 😀 The domain of the function is the set {1, 2, 3, 4, 5}, as these are the elements of set A.
• 😀 The codomain of the function is the set {1, 2, 3, 4, 5, 6}, as these are all the elements of set B, regardless of whether they are actually mapped from A.
• 😀 The image of the function is the set {2, 3, 4, 5, 6}, as these are the elements in B that actually received arrows from A in the diagram.
• 😀 The function's rule for this example was f(x) = x + 1, meaning each value in A was mapped to a value in B by adding 1 to the element.
• 😀 The transcript also demonstrated how to reverse the process when given the image, working through exercises to determine the function and domain by using algebraic manipulation.

Q & A

What is the domain of a function?

The domain of a function refers to the set of all possible input values (elements of set A) that the function can accept.

How do you define the codomain of a function?

The codomain of a function is the set of all possible output values (elements of set B) that the function can produce, regardless of whether every element in the codomain is actually mapped to by the function.

What is the image of a function?

The image of a function is the set of actual output values that are produced by applying the function to the elements of the domain. These are the elements in the codomain that are actually mapped to by the function.

Can you explain the relationship between the domain, codomain, and image of a function using the example provided?

In the example, the domain (set A) consists of {1, 2, 3, 4, 5}. The codomain (set B) consists of {1, 2, 3, 4, 5, 6, 7}. The image of the function, which consists of the elements of set B that are actually mapped to by the function, is {2, 3, 4, 5, 6}.

What role do the arrows play in the function example given?

The arrows represent the relationship between the elements of the domain and the codomain. Each arrow indicates which element of the codomain corresponds to an element of the domain according to the function.

How do you determine the domain, codomain, and image from a diagram of arrows?

To determine the domain, identify all the elements from the set A (inputs). To find the codomain, look at all the possible output elements in set B. To identify the image, check which elements in the codomain actually have arrows pointing to them from the domain.

In the example, how is the function defined?

In the example, the function is defined as f(x) = x + 1, meaning each element of the domain (set A) is mapped to an element of the codomain (set B) by adding 1 to it.

Why is the element '1' not part of the image in the given example?

The element '1' is not part of the image because it does not receive any arrows from the domain in the function. Only the elements that actually have arrows pointing to them are part of the image.

What is the difference between codomain and image in terms of their content?

The codomain includes all possible outputs, whether they are actually produced by the function or not. The image, on the other hand, only includes those outputs that the function actually maps to from the domain.

How would you solve an exercise to find the domain, codomain, and image of a function when given a set of values?

To solve such an exercise, first identify the domain by looking at the set of input values. Then determine the codomain by examining the potential output values. Finally, find the image by identifying the actual outputs based on the given function and any relationships indicated by the diagram of arrows.