Example Question - vertical asymptote
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Understanding a Function's Graph
La imagen muestra la gráfica de una función f(x) que está definida como: \[ f(x) = \frac{x}{x^2 - 8x + 16} \] Para comprender la gráfica, primero debemos simplificar y entender la expresión algebraica dada. Observa que el denominador \( x^2 - 8x + 16 \) es un trinomio cuadrado perfecto que se puede factorizar como \((x - 4)^2\). Así que la función se puede reescribir como: \[ f(x) = \frac{x}{(x - 4)^2} \] Dado que la gráfica de la función ya está dada en la imagen, describiremos las características más importantes: 1. La función tiene una asíntota vertical en x = 4, que es el valor para el cual el denominador se hace cero y la función se indefine. En el gráfico se observa que la función crece indefinidamente a medida que x se aproxima a 4 desde la izquierda y decrece indefinidamente a medida que x se aproxima a 4 desde la derecha. 2. La función tiene una asíntota horizontal en y = 0 cuando x tiende a infinito o a menos infinito, ya que el grado del polinomio en el denominador es mayor que el grado del numerador. La gráfica se aproxima a la línea y = 0 pero nunca la toca. 3. La función cruza el eje de las ordenadas (eje y) en el punto (0, 0), ya que f(0) = 0. 4. La función es positiva para valores de x menores que 4 y mayores que 4, esto es reflejado por el hecho de que la gráfica está por encima del eje x en esos intervalos. 5. La función es simétrica respecto al eje y porque el numerador es x (una función impar) y el denominador es una función par ((x - 4)^2). Esto resulta en una función impar, lo que explica por qué la gráfica es simétrica respecto al origen. Si tienes alguna otra pregunta específica sobre la gráfica o la función, no dudes en preguntar.
Logarithmic Function Transformation Analysis
The given logarithmic function f(x) = log(x) is transformed to g(x) = log(x + 1) + 3. To determine which of the statements is true, we need to analyze the transformation that g(x) represents compared to f(x). - For statement A: f(x) is translated 1 unit upward. This is not true because the function g(x) has a "+3" outside of the logarithm, which means the entire function is translated 3 units upward, not 1 unit. - For statement B: f(x) is translated 3 units downward. This is incorrect as the transformation involves a "+3", indicating an upward translation, not a downward one. - For statement C: The vertical asymptote shifts 1 unit to the left. This is true. The vertical asymptote for the basic logarithmic function f(x) = log(x) is at x=0. Since g(x) is log(x + 1), this transformation shifts the graph horizontally 1 unit to the left, meaning the new vertical asymptote is now at x=-1. - For statement D: The vertical asymptote shifts 3 units to the right. This is not true. The transformation inside the logarithmic function (x + 1) does not shift the graph to the right; it shifts it to the left. Hence, the correct statement is C: The vertical asymptote shifts 1 unit to the left.
Limit Calculation on Function Graph
The image shows a function graphed on a coordinate plane, and there are several limits that need to be calculated. While I cannot interact with the image to enlarge it or click on it, I can analyze it based on the image provided. 1. The first limit is \(\lim_{x\to c^-} f(x)\), where `c` is the x-value at the vertical asymptote where the graph tends towards infinity as x approaches from the left. From the image, the graph shoots upwards towards positive infinity as x approaches this value from the left side. 2. The second limit is \(\lim_{x\to c^+} f(x)\), for the same value `c` (the vertical asymptote). From the graph, when approaching from the right side of the vertical asymptote, the function appears to head downwards towards negative infinity. 3. The third limit is \(\lim_{x\to d^-} f(x)\), where `d` represents the x-value where the function is heading toward as x approaches from the left. In this case, the graph approaches a horizontal asymptote from the left. However, without precise values or markings on the graph, I cannot indicate what the value of the limit is, so I can generally say that the limit has a certain finite value, which we can call 'L', where \(L\) is the y-value of the horizontal asymptote. 4. The fourth limit is \(\lim_{x\to d^+} f(x)\), which is as x approaches `d` from the right. From the graph, it seems the function approaches the same horizontal asymptote as it does from the left, thus also appearing to approach the same value 'L'. To summarize, the answers based on the graph (using general terms since specific values are not given): 1. \(\lim_{x\to c^-} f(x) = \infty\) 2. \(\lim_{x\to c^+} f(x) = -\infty\) 3. \(\lim_{x\to d^-} f(x) = L\) 4. \(\lim_{x\to d^+} f(x) = L\) Please bear in mind that 'L' stands for the finite limit value at the horizontal asymptote. In a precise solution, you would need to replace this with the actual y-coordinate value of the horizontal asymptote.
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