FUNÇÃO INVERSA | DEFINIÇÃO, LEI DE FORMAÇÃO, DIAGRAMA E EXERCÍCIOS | Ensino médio - Gis com Giz

Summary

In this educational video, the instructor uses a creative approach to explain the concept of inverse functions in mathematics. Using a hamburger as a metaphor, the instructor demonstrates how the elements of a function can be 'inverted' by switching their places, much like swapping the bread and patty of a sandwich. The lesson covers defining a function, calculating its inverse, and understanding the relationship between the domain, codomain, and image. Throughout, the instructor provides clear examples, simplifying the process of understanding inverse functions, while also emphasizing the importance of grasping the fundamental concepts.

Takeaways

• 😀 The lesson introduces the concept of inverse functions using an engaging, real-life analogy with a hamburger.
• 😀 The function is explained through a simple mathematical formula: f(x) = x + 2.
• 😀 The relationship between the domain and codomain is demonstrated using sets: set A (1, 2, 3) and set B (3, 4, 5).
• 😀 The function f(x) = x + 2 establishes a correspondence between elements in the domain and codomain.
• 😀 The script emphasizes how to identify and write ordered pairs for a function: (1, 3), (2, 4), (3, 5).
• 😀 The concept of an inverse function is introduced by switching the order of elements in the domain and codomain.
• 😀 The inverse function of f(x) = x + 2 is found by switching x and y and solving for y.
• 😀 To calculate the inverse function, you isolate y after swapping the roles of x and y, resulting in y = x - 2.
• 😀 The lesson also demonstrates a faster way to calculate the inverse of a function without drawing diagrams, providing a more efficient method.
• 😀 Several examples are worked through to solidify understanding, such as calculating f^(-1)(4) for the function f(x) = 3x - 2.

Q & A

What is an inverse function?

An inverse function is a function that reverses the operation of the original function, swapping the roles of the input and output. For example, if a function adds 2 to an input, its inverse will subtract 2 from the output.

How does the analogy of a hamburger help in understanding inverse functions?

The hamburger analogy helps visualize how the input and output of a function are swapped in the inverse function. In the analogy, swapping the bread and patty represents swapping the domain and range of the function.

How do you calculate an inverse function step by step?

To calculate an inverse function, first replace the function notation (f(x)) with y. Then, swap x and y, and solve for y. The resulting equation will give the inverse function.

What is the inverse of the function f(x) = x + 2?

The inverse of the function f(x) = x + 2 is f⁻¹(x) = x - 2. This is found by swapping x and y, then solving for y.

Why do we swap x and y when finding the inverse of a function?

We swap x and y because the inverse function reverses the mapping of the original function, switching the roles of input and output.

What are the domain and range of the function f(x) = x + 2?

In the function f(x) = x + 2, the domain consists of the input values (1, 2, 3), and the range consists of the corresponding output values (3, 4, 5).

What is the role of the law of formation in understanding functions?

The law of formation defines how each element of the domain is mapped to an element in the range. In the case of f(x) = x + 2, it shows that each input x is mapped to x + 2.

How do you use the inverse function to find f⁻¹(4) for the function f(x) = 3x - 2?

To find f⁻¹(4) for f(x) = 3x - 2, first calculate the inverse function, which is f⁻¹(x) = (x + 2) / 3. Then, substitute 4 for x, yielding f⁻¹(4) = (4 + 2) / 3 = 6 / 3 = 2.

What is the difference between the function f(x) = x + 2 and its inverse function f⁻¹(x) = x - 2?

The function f(x) = x + 2 adds 2 to the input, whereas the inverse function f⁻¹(x) = x - 2 subtracts 2 from the output. These two functions reverse the effect of each other.

How does the 'sandwich' analogy connect to the concept of inverse functions?

The 'sandwich' analogy is used to illustrate how the order of elements is swapped when finding the inverse function. Just like swapping the bread and patty in a sandwich, the input and output of a function are swapped in the inverse function.