Example Question - function intervals
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Piecewise Function Analysis
La función mostrada en la imagen es una función a trozos, que tiene una forma diferente dependiendo del intervalo en el que se encuentre la variable independiente \( x \). La definición de la función es la siguiente: \[ f(x) = \begin{cases} 0.5x^2 + 2 & \text{si } x < 4 \\ 2x + 2 & \text{si } 4 \leq x < 10 \\ 7 & \text{si } x \geq 10 \end{cases} \] Para resolver la función para un valor específico de \( x \), necesitas determinar primero en qué intervalo se encuentra ese valor y luego usar la expresión correspondiente. Si por ejemplo queremos calcular el valor de la función para \( x = 3 \), ya que \( 3 < 4 \), usaríamos la primera expresión: \[ f(3) = 0.5 \cdot 3^2 + 2 = 0.5 \cdot 9 + 2 = 4.5 + 2 = 6.5 \] Para calcular el valor de la función para \( x = 5 \), ya que \( 4 \leq 5 < 10 \), usaríamos la segunda expresión: \[ f(5) = 2 \cdot 5 + 2 = 10 + 2 = 12 \] Y finalmente, para calcular el valor para \( x = 11 \), como \( 11 \geq 10 \), usaríamos la tercera expresión que es una constante: \[ f(11) = 7 \] Puedes usar este método para encontrar el valor de la función para cualquier valor de \( x \).
Identifying Function Increase and Decrease Intervals
The image displays the graph of a function, and you are asked to determine the intervals where the function is increasing or decreasing. When a function is increasing, the y-values (the values on the vertical axis) get larger as the x-values (the values on the horizontal axis) increase. Conversely, a function is decreasing when the y-values get smaller as the x-values increase. Looking at the graph: - The function is increasing from approximately x=-4 to x=-2.5, x=-1 to x=0, and x=2 to x=3. - The function is decreasing from x=-5 to x=-4, x=-2.5 to x=-1, x=0 to x=2, and x=3 to the end of the graph at x=4. To describe these intervals, you can use interval notation, where you use parentheses to denote values that are not included in the interval (open interval) and square brackets for values that are included (closed interval). From the graph, it is not clear whether the extrema (the highest and lowest points along each interval) are included because we do not have precise points. However, since extrema are usually considered part of both increasing and decreasing intervals, if the interval is described using precise points, such points are typically included. Therefore, the intervals should be described as roughly: - Increasing: (-4, -2.5), (-1, 0), and (2, 3) - Decreasing: (-5, -4), (-2.5, -1), (0, 2), and (3, 4) If these values were precisely known from the graph, closed brackets would be used at the extrema. It is important to remember that the exact values may differ slightly, and it's often best to get these values from the context of the question or the accompanying material.
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