Pregunta
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For the given functions \( f \) and \( g \), complete parts \( (\mathrm{a})-(\mathrm{h}) \) For parts \( (\mathrm{a}) \)-(d), also find the domain. \( f(\mathrm{x})=\mathrm{x}-1 ; g(\mathrm{x})=4 x^{2} \)

Ask by Garrett Pope. in the United States
Mar 11,2025

Solución de inteligencia artificial de Upstudy

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Parts (a) through (d) are as follows: - **(a)** \( f(g(x)) = 4x^2 - 1 \), Domain: All real numbers. - **(b)** \( g(f(x)) = 4(x - 1)^2 \), Domain: All real numbers. - **(c)** \( f(x) + g(x) = 4x^2 + x - 1 \), Domain: All real numbers. - **(d)** \( f(x) \cdot g(x) = 4x^3 - 4x^2 \), Domain: All real numbers. Please specify parts (e) through (h) for further analysis.

Solución

Let's analyze the functions \( f(x) = x - 1 \) and \( g(x) = 4x^2 \) and complete the required parts step by step. ### Part (a): Find \( f(g(x)) \) To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(4x^2) = 4x^2 - 1 \] ### Domain of \( f(g(x)) \) The function \( g(x) = 4x^2 \) is defined for all real numbers \( x \). Since \( f(x) \) is also defined for all real numbers, the domain of \( f(g(x)) \) is: \[ \text{Domain of } f(g(x)) = (-\infty, \infty) \] ### Part (b): Find \( g(f(x)) \) To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(x - 1) = 4(x - 1)^2 \] ### Domain of \( g(f(x)) \) The function \( f(x) = x - 1 \) is defined for all real numbers \( x \). Since \( g(x) \) is also defined for all real numbers, the domain of \( g(f(x)) \) is: \[ \text{Domain of } g(f(x)) = (-\infty, \infty) \] ### Part (c): Find \( f(x) + g(x) \) To find \( f(x) + g(x) \): \[ f(x) + g(x) = (x - 1) + (4x^2) = 4x^2 + x - 1 \] ### Domain of \( f(x) + g(x) \) Both \( f(x) \) and \( g(x) \) are defined for all real numbers, so the domain of \( f(x) + g(x) \) is: \[ \text{Domain of } f(x) + g(x) = (-\infty, \infty) \] ### Part (d): Find \( f(x) \cdot g(x) \) To find \( f(x) \cdot g(x) \): \[ f(x) \cdot g(x) = (x - 1)(4x^2) = 4x^3 - 4x^2 \] ### Domain of \( f(x) \cdot g(x) \) Both functions are defined for all real numbers, so the domain of \( f(x) \cdot g(x) \) is: \[ \text{Domain of } f(x) \cdot g(x) = (-\infty, \infty) \] ### Summary of Parts (a)-(d) - **(a)** \( f(g(x)) = 4x^2 - 1 \), Domain: \( (-\infty, \infty) \) - **(b)** \( g(f(x)) = 4(x - 1)^2 \), Domain: \( (-\infty, \infty) \) - **(c)** \( f(x) + g(x) = 4x^2 + x - 1 \), Domain: \( (-\infty, \infty) \) - **(d)** \( f(x) \cdot g(x) = 4x^3 - 4x^2 \), Domain: \( (-\infty, \infty) \) Now, please specify parts \( (\mathrm{e}) - (\mathrm{h}) \) for further analysis.

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The Deep Dive

To complete parts \( (\mathrm{a})-(\mathrm{h}) \), we first start by understanding the functions given. The function \( f(x) = x - 1 \) is a linear function, and its domain is all real numbers, \( \mathbb{R} \). The function \( g(x) = 4x^2 \) is a quadratic function, and its domain is also all real numbers, \( \mathbb{R} \). **(a) Domain of \( f \)**: The domain of \( f(x) \) is \( \mathbb{R} \) (all real numbers). **(b) Domain of \( g \)**: The domain of \( g(x) \) is again \( \mathbb{R} \) (all real numbers). **(c) \( f(g(x)) \)**: To find \( f(g(x)) \), substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(4x^2) = 4x^2 - 1. \] The domain of \( f(g(x)) \) is still \( \mathbb{R} \). **(d) \( g(f(x)) \)**: To find \( g(f(x)) \), substitute \( f(x) \) into \( g \): \[ g(f(x)) = g(x - 1) = 4(x - 1)^2 = 4(x^2 - 2x + 1) = 4x^2 - 8x + 4. \] The domain of \( g(f(x)) \) remains \( \mathbb{R} \). Now let's move on to parts (e) to (h) which could involve various operations such as addition, multiplication, composition, or finding specific values. **(e) \( f(x) + g(x) \)**: \[ f(x) + g(x) = (x - 1) + (4x^2) = 4x^2 + x - 1. \] Domain: \( \mathbb{R} \). **(f) \( f(x) - g(x) \)**: \[ f(x) - g(x) = (x - 1) - (4x^2) = -4x^2 + x - 1. \] Domain: \( \mathbb{R} \). **(g) \( f(x) \cdot g(x) \)**: \[ f(x) \cdot g(x) = (x - 1)(4x^2) = 4x^3 - 4x^2. \] Domain: \( \mathbb{R} \). **(h) \( \frac{f(x)}{g(x)} \)**: \[ \frac{f(x)}{g(x)} = \frac{x - 1}{4x^2} = \frac{1}{4} \cdot \frac{x - 1}{x^2}. \] Domain: \( x \neq 0 \) (cannot divide by zero). So there you have it! Functions, their domains, and their combinations all rolled into one neat package. Happy function exploring!

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