FUNCIÓN VALOR ABSOLUTO Análisis Completo: Vértice, dominio, rango, ceros función, intervalos C+ y C-

Summary

In this video, the presenter explains the complete analysis of the absolute value function f(x) = |x - 3| - 2. The content covers key concepts such as the graph's shape depending on the sign of the function, the vertex, and how to find the zeros (roots) of the function. It also discusses finding the y-intercept and the domain and range of the function. The video further explains intervals of positivity, negativity, and the function's behavior, including its growth and decay. Viewers will gain a clear understanding of the graphical representation and properties of the absolute value function.

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Q & A

What is the main function being analyzed in this video?

The main function being analyzed is the absolute value function: f(x) = |x - 3| - 2.

How does the sign in front of the absolute value function affect its graph?

The sign in front of the absolute value function determines the orientation of the graph. If the sign is positive, the graph will have a 'V' shape, while a negative sign causes the graph to take an inverted 'V' shape.

What is the vertex of the given absolute value function?

The vertex of the function is the point (3, -2). This is found by solving for x in the expression inside the absolute value and then calculating the corresponding y value.

How do you calculate the x-coordinate of the vertex for this function?

The x-coordinate of the vertex is found by setting the expression inside the absolute value to zero: x - 3 = 0, which gives x = 3.

What is the y-coordinate of the vertex for this function?

The y-coordinate is found by substituting x = 3 into the function, which results in f(3) = |3 - 3| - 2 = 0 - 2 = -2. Thus, the vertex is at (3, -2).

How are the zeros (roots) of the function determined?

To find the zeros of the function, set f(x) = 0 and solve the equation |x - 3| - 2 = 0. This results in two possible solutions: x = 1 and x = 5.

What is the process for calculating the y-intercept (the order of the origin)?

To calculate the y-intercept, set x = 0 in the function. Substituting x = 0 into f(x) = |x - 3| - 2 gives f(0) = |0 - 3| - 2 = 3 - 2 = 1. Therefore, the y-intercept is 1.

What is the domain of this function?

The domain of the function is all real numbers, as there are no restrictions on the values that x can take. This is represented as (-∞, ∞).

What is the range of this function?

The range of the function is all values greater than or equal to -2, since the vertex represents the minimum value of the function. The range is expressed as [-2, ∞).

What are the intervals where the function is positive and negative?

The function is positive on the intervals (-∞, 1) and (5, ∞), and negative on the interval (1, 5).