Pregunta
upstudy study bank question image url

2) Consider the function \( f(x)=2 x^{2} \) and \( p(x)=\left(\frac{1}{2}\right)^{x} \). a) Restrict the domain of \( f \) so that the inverse of \( f \) will also be a function. b) Write down the equation of the inverse of \( p \) in the form \( p^{-1}(x)=\ldots \) c) If the inverse of another function is \( g^{-1}(x)=2 x-4 \). Determine the equation of \( g(x) \).

Ask by Gray Ortega. in South Africa
Feb 20,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

a) Restrict the domain of \( f(x) = 2x^2 \) to \( x \geq 0 \). b) The inverse of \( p(x) = \left(\frac{1}{2}\right)^x \) is \( p^{-1}(x) = \log_{\frac{1}{2}}(x) \). c) If \( g^{-1}(x) = 2x - 4 \), then \( g(x) = \frac{x + 4}{2} \).

Solución

We are given the functions f(x) = 2x² and p(x) = (1/2)ˣ. We wish to: a) Restrict the domain of f so its inverse is a function. b) Find the inverse of p in the form p⁻¹(x) = … c) Given g⁻¹(x) = 2x – 4, find g(x). Let’s work through each part step by step. ──────────────────────── Part (a) For a function to have an inverse that is also a function, the function must be one-to-one (injective). The function f(x) = 2x² is not one-to-one over the real numbers because, for example, f(2) = 8 and f(–2) = 8. A common remedy is to restrict the domain so that the function is either only increasing or only decreasing. A common restriction is to choose x ≥ 0, which gives a monotonically increasing function. Thus, by restricting the domain to [0, ∞), f becomes one-to-one and its inverse is defined. ──────────────────────── Part (b) We wish to find the inverse of p(x) = (1/2)ˣ. The steps are: 1. Begin with y = (1/2)ˣ. 2. In order to solve for x, take the logarithm base (1/2) on both sides:   x = log₍₁/₂₎(y). Since we want the inverse function written as p⁻¹(x), we interchange x and y:   p⁻¹(x) = log₍₁/₂₎(x). Thus, the inverse is:   p⁻¹(x) = log₍₁/₂₎(x). ──────────────────────── Part (c) We are given that g⁻¹(x) = 2x – 4 and need to find the corresponding function g(x). Recall that g and g⁻¹ are inverses, so:   g(g⁻¹(x)) = x. However, a more straightforward way is to “invert” the function g⁻¹(x). Write   y = 2x – 4 and solve for x in terms of y. Step 1. Start with:   y = 2x – 4 Step 2. Add 4 to both sides:   y + 4 = 2x Step 3. Divide both sides by 2:   x = (y + 4)/2 Now, reinterpret this equation. Since we began with y = g⁻¹(x) and ended with x in terms of y, the inverse of g⁻¹ is the function g. Renaming y back as the variable x, we get:   g(x) = (x + 4)/2. ──────────────────────── Final Answers: a) Restrict the domain of f to x ≥ 0. b) p⁻¹(x) = log₍₁/₂₎(x). c) g(x) = (x + 4)/2.

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Beyond the Answer

To ensure \( f(x) = 2x^2 \) has an inverse that is also a function, we should restrict its domain to either \( x \geq 0 \) or \( x \leq 0 \). For instance, choosing \( x \geq 0 \) makes \( f \) a one-to-one function, allowing us to find a clean inverse without ambiguity in y-values. For the function \( p(x) = \left(\frac{1}{2}\right)^{x} \), we can find the inverse by solving for \( x \) in terms of \( y \). Setting \( y = p(x) \) gives \( y = \left(\frac{1}{2}\right)^{x} \). Taking the logarithm, we find \( x = -\log_{1/2}(y) \), giving us the inverse as \( p^{-1}(x) = -\log_{1/2}(x) \).

Latest Pre Calculus Questions

¡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