Example Question - range
Here are examples of questions we've helped users solve.
Graphing Intervals and Functions
<p>Para representar gráficamente los intervalos, comenzamos con:</p> <p>a. [1, 2): Una línea sólida desde 1 hasta 2, sin incluir 2.</p> <p>b. (-∞, 2]: Una línea sólida que se extiende desde -∞ hasta 2, incluyendo 2.</p> <p>c. (-5, 0): Una línea sólida desde -5 hasta 0, sin incluir 0.</p> <p>d. [2, ∞): Una línea sólida desde 2 hacia el infinito, incluyendo 2.</p> <p>e. (-∞, -1]: Una línea sólida que se extiende desde -∞ hasta -1, incluyendo -1.</p> <p>Para la función:</p> <p>a. \( y = \sqrt{x - 4} \): El dominio es \( x \geq 4 \) y el rango es \( y \geq 0 \).</p> <p>b. \( y = \sqrt{x + 1} \): El dominio es \( x \geq -1 \) y el rango es \( y \geq 0 \).</p>
Finding the Domain and Range of a Given Function
<p>The given function is \( f(x) = \frac{1}{\sqrt{9-x^2}} \).</p> <p>To find the domain, we need to identify all values of \( x \) for which the function is defined. The denominator cannot be zero, and the expression under the square root must be non-negative because the root must be a real number.</p> <p>We have the inequality \( 9 - x^2 \geq 0 \) which leads to \( -3 \leq x \leq 3 \).</p> <p>Hence, the domain of \( f(x) \) is \( [-3, 3] \).</p> <p>To find the range, we know that the function is always positive because it is the reciprocal of the square root of a positive number, and the square root function only returns non-negative values.</p> <p>Since the square root function \( \sqrt{9-x^2} \) will have values ranging from \( 0 \) (not included because it would make the denominator zero) to \( 3 \), the reciprocal of this function will range from \( \frac{1}{3} \) to infinity.</p> <p>Hence, the range of \( f(x) \) is \( (0, \infty) \).</p>
Listing Integers in a Specific Range
<p>The question requires listing all integers from \(-3\) to \(3\).</p> <p>\(-3, -2, -1, 0, 1, 2, 3\)</p> <p>Therefore, the correct answer is option B.</p>
Domain and Range Determination for Given Functions
<p>Untuk soal (a):</p> <p>D_{f(x,y)} = \{(x, y) \in \mathbb{R}^2 | x^2 - y^2 - 9 \geq 0\}</p> <p>R_{f(x,y)} = \{f(x,y) \in \mathbb{R} | f(x,y) = 4-x^2-y^2 + \frac{1}{2}\log(x^2 - y^2 - 9), (x, y) \in D_{f(x,y)} \}</p> <p>Untuk soal (b):</p> <p>D_{f(x,y)} = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 - 1 \neq 0\}</p> <p>R_{f(x,y)} = \{f(x,y) \in \mathbb{R} | f(x,y) = \frac{\sqrt{x^2 - y^2}}{x^2 + y^2 - 1}, (x, y) \in D_{f(x,y)} \}</p>
Analysis of Parabola Characteristics
La imagen proporciona dos funciones cuadráticas para analizar sus características. Vamos a determinar estas características para la primera función \( f(x) = 2x^2 - 6x - 4 \). <p>Para encontrar la orientación de la parábola, observamos el coeficiente líder \(a\).</p> <p>Si \( a > 0 \), la parábola se abre hacia arriba. Si \( a < 0 \), se abre hacia abajo.</p> <p>En nuestro caso, \( a = 2 \), por lo que la parábola se abre hacia arriba.</p> <p>El eje de simetría de una parábola se encuentra en \( x = -\frac{b}{2a} \).</p> <p>Sustituimos \( a = 2 \), \( b = -6 \) para obtener \( x = -\frac{-6}{2 \cdot 2} = \frac{3}{2} \).</p> <p>El vértice de la parábola se encuentra en el punto \( (\frac{-b}{2a}, f(\frac{-b}{2a})) \).</p> <p>Calculamos \( f(\frac{3}{2}) = 2(\frac{3}{2})^2 - 6(\frac{3}{2}) - 4 \).</p> <p>El vértice es \( (\frac{3}{2}, -\frac{25}{4}) \).</p> <p>El intercepto en x son los ceros de la función, donde \( f(x) = 0 \).</p> <p>Resolvemos \( 2x^2 - 6x - 4 = 0 \) usando la fórmula cuadrática o factorización para encontrar los ceros.</p> <p>El intercepto en y es \( f(0) \), es decir, \( -4 \).</p> <p>El dominio de cualquier función cuadrática es \( (-\infty, \infty) \).</p> <p>El recorrido (rango) depende de la orientación de la parábola. Como se abre hacia arriba, el rango es \( [f(\frac{-b}{2a}), \infty) \), es decir, \( [-\frac{25}{4}, \infty) \).</p> Este es un análisis completo para la primera función. Para la segunda función, se seguiría un proceso similar.
Calculating the Range of a Series of Numbers
<p>To find the range of the given data set, identify the largest and smallest numbers and subtract the smallest from the largest.</p> <p>Smallest number = \(11\)</p> <p>Largest number = \(71\)</p> <p>\( \text{Range} = \text{Largest number} - \text{Smallest number} = 71 - 11 = 60 \)</p>
Calculating the Range of a Set of Numbers
<p>To find the range of the data set, we subtract the smallest number from the largest number.</p> <p>The smallest number in the set is \(11\).</p> <p>The largest number in the set is \(71\).</p> <p>Therefore, the range is \(71 - 11 = 60\).</p>
Determining the Domain and Range from a Graph
<p>The graph shows a function with two distinct parts. The first part is decreasing and the second part is increasing. There is a break in the graph where the function is not defined.</p> <p>To find the domain, we look for the x-values that the function covers. By observing the graph, we see that the function is defined for all x except for a portion where x is between -4 and 3. Thus the domain is \( x < -4 \) or \( x > 3 \).</p> <p>The range of a function is the set of all possible output values (y-values), which result from using the function's formula. By examining the graph, we see that as \( x \) approaches -4 from the left, the y-values decrease without bound, and as \( x \) approaches 3 from the right, the y-values increase without bound. Therefore, the range of the function is all real numbers, which can be denoted as \( -\infty < y < \infty \) or simply \( y \in \mathbb{R} \).</p>
Determining the Domain and Range from a Graph
\[ \text{Domain: } \{ x \mid x < -4 \text{ or } x \geq 4\} \] \[ \text{Range: } \{ y \mid y \geq 4 \} \]
Identifying the Range of an Inequality
<p>Verilen eşitsizlikteki sayılar arasındaki mesafeyi ve aralığı bulmamız gerekiyor.</p> <p>Bir sayı çizelgesi üzerinde, -5 ile 0 arasındaki her sayı bu eşitsizliği sağlamaktadır. </p> <p>Eşitsizlik \(-5 < x < 0\) şeklinde ifade edilebilir. </p> <p>Yani sayı çizgisi üzerindeki açık parantezler (-5) ve (0) arasındaki sayıları kapsar. </p>
Finding Multiples of 5 in a Range
To solve this problem, we need to find the first and the last integers between 1 and 100 that are divisible by 5, and then determine how many such integers there are in total. The smallest multiple of 5 between 1 and 100 is 5 itself, and the largest multiple of 5 is 100 (since 100 is a multiple of 5). To find the number of multiples of 5 from 5 to 100, we can use the following formula: Number of multiples of 5 = (Last multiple of 5 - First multiple of 5) / 5 + 1 Substituting the appropriate values we get: Number of multiples of 5 = (100 - 5) / 5 + 1 Number of multiples of 5 = 95 / 5 + 1 Number of multiples of 5 = 19 + 1 Number of multiples of 5 = 20 There are 20 integers between 1 and 100 that are divisible by 5. The correct answer is option C) 20.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com