Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Question
### Functions and Their Domains The functions \( f \) and \( g \) are defined as: \[ f(x) = 4x + 3 \] \[ g(x) = 1 - 2x \] #### Question a) Find the domain of the following expressions: \( f \), \( g \), \( f + g \), \( f - g \), \( fg \), \( \frac{f}{g} \), and \( \frac{g}{f} \). #### Question b) Find the following compositions and products of functions: \[ (f + g)(x) \] \[ (f - g)(x) \] \[ (fg)(x) \] \[ (ff)(x) \] \[ \left(\frac{f}{g}\right)(x) \] \[ \left(\frac{g}{f}\right)(x) \] #### Answer Box for Question a) The domain of \( f \) is: \[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \] (Type your answer in interval notation.) --- ### Explanations - **Function Definitions**: - \( f(x) = 4x + 3 \) is a linear function. - \( g(x) = 1 - 2x \) is also a linear function. - **Domain of a Function**: - The domain of a linear function like \( f(x) \) or \( g(x) \) is typically all real numbers, unless otherwise restricted. To determine the domain of compositions and products of these functions, or when dealing with ratios \( \frac{f}{g} \) and \( \frac{g}{f} \), consider where the functions are undefined (such as divisions by zero). For \( \frac{f}{g} \): \[ \frac{4x + 3}{1 - 2x} \] The function is undefined when \( g(x) = 0 \): \[ 1 - 2x = 0 \Rightarrow x = \frac{1}{2} \] Thus, the function \( \frac{f}{g} \) is undefined at \( x = \frac{1}{2} \). For \( \frac{g}{f} \): \[ \frac{1 - 2x}{4x + 3} \] The function is
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Transcribed Image Text:### Functions and Their Domains The functions \( f \) and \( g \) are defined as: \[ f(x) = 4x + 3 \] \[ g(x) = 1 - 2x \] #### Question a) Find the domain of the following expressions: \( f \), \( g \), \( f + g \), \( f - g \), \( fg \), \( \frac{f}{g} \), and \( \frac{g}{f} \). #### Question b) Find the following compositions and products of functions: \[ (f + g)(x) \] \[ (f - g)(x) \] \[ (fg)(x) \] \[ (ff)(x) \] \[ \left(\frac{f}{g}\right)(x) \] \[ \left(\frac{g}{f}\right)(x) \] #### Answer Box for Question a) The domain of \( f \) is: \[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \] (Type your answer in interval notation.) --- ### Explanations - **Function Definitions**: - \( f(x) = 4x + 3 \) is a linear function. - \( g(x) = 1 - 2x \) is also a linear function. - **Domain of a Function**: - The domain of a linear function like \( f(x) \) or \( g(x) \) is typically all real numbers, unless otherwise restricted. To determine the domain of compositions and products of these functions, or when dealing with ratios \( \frac{f}{g} \) and \( \frac{g}{f} \), consider where the functions are undefined (such as divisions by zero). For \( \frac{f}{g} \): \[ \frac{4x + 3}{1 - 2x} \] The function is undefined when \( g(x) = 0 \): \[ 1 - 2x = 0 \Rightarrow x = \frac{1}{2} \] Thus, the function \( \frac{f}{g} \) is undefined at \( x = \frac{1}{2} \). For \( \frac{g}{f} \): \[ \frac{1 - 2x}{4x + 3} \] The function is
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