Pregunta
upstudy study bank question image url

Sketch a rational function subject to the given conditions. Horizontal asymptote: \( y=2 \) Vertical asymptote: \( x=1 \) \( y \)-intercept: \( (0,3) \) \( x \)-intercept: \( \left(\frac{3}{2}, 0\right) \)

Ask by Byrd Murphy. in the United States
Mar 18,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The rational function that satisfies all the given conditions is: \[ f(x) = \frac{2x - 3}{x - 1} \]

Solución

Consider a rational function of the form \[ f(x)=\frac{p(x)}{q(x)} \] where we choose \( p(x) \) and \( q(x) \) such that all conditions are satisfied. **Step 1. Horizontal asymptote: \( y=2 \)** For the horizontal asymptote to be \( y=2 \), the degrees of the numerator and denominator must be the same and the ratio of their leading coefficients must be 2. A natural choice is to let both the numerator and denominator be linear. Let \[ p(x)=2x+b \quad \text{and} \quad q(x)=x-1. \] The leading coefficient of the numerator is 2 and that of the denominator is 1, so \[ \lim_{x \to \infty} f(x)=\frac{2}{1}=2. \] This satisfies the horizontal asymptote condition. **Step 2. Vertical asymptote: \( x=1 \)** For \( x=1 \) to be a vertical asymptote, the denominator must be zero at \( x=1 \) (and not cancel with the numerator). With \[ q(x)=x-1, \] we have \( q(1)=0 \). Therefore, the function has a vertical asymptote at \( x=1 \). **Step 3. \( y \)-intercept: \( (0,3) \)** The \( y \)-intercept is given when \( x=0 \): \[ f(0)=\frac{2(0)+b}{0-1}=\frac{b}{-1}=-b. \] We need \[ -b=3 \quad \Rightarrow \quad b=-3. \] Thus, the numerator becomes \[ p(x)=2x-3. \] **Step 4. \( x \)-intercept: \( \left(\frac{3}{2},0\right) \)** The \( x \)-intercept is found by setting the numerator equal to zero: \[ 2x-3=0 \quad \Rightarrow \quad 2x=3 \quad \Rightarrow \quad x=\frac{3}{2}. \] This matches the given \( x \)-intercept. **Final Function** The function that meets all the conditions is \[ f(x)=\frac{2x-3}{x-1}. \]

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

The Deep Dive

To sketch a rational function that meets the specified conditions, consider a function of the form \( f(x) = \frac{a(x - \frac{3}{2})}{x - 1} \), where \( a \) will be adjusted to ensure the horizontal asymptote and \( y \)-intercept align with the requirements. Given that there’s a horizontal asymptote at \( y = 2 \), the leading coefficients of the numerator and denominator should be \( 2 \) and \( 1 \) respectively. So you can structure the function as: \[ f(x) = \frac{2(x - \frac{3}{2})}{x - 1} \] Next, simplify this to: \[ f(x) = \frac{2x - 3}{x - 1} \] Now, check the key points: - **Horizontal asymptote \( y=2 \)**: As \( x \) approaches infinity, the \( f(x) \approx \frac{2x}{x} = 2 \). - **Vertical asymptote \( x=1 \)**: \( f(x) \) becomes undefined when \( x=1 \). - **\( y \)-intercept \( (0,3) \)**: Substitute \( x=0 \): \[ f(0) = \frac{2(0) - 3}{0 - 1} = \frac{-3}{-1} = 3 \] - **\( x \)-intercept \( \left(\frac{3}{2}, 0\right) \)**: Set the numerator equal to zero: \[ 2x - 3 = 0 \rightarrow x = \frac{3}{2} \] Now, plot these key features on a graph to visualize the function, keeping in mind the asymptotic behavior near \( x=1 \) and the intercepts.

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad