Questions: For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x)=x-8 ; g(x)=2x^2 What is the domain of f-g? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is x . (Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The domain is x x is any real number .

$For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x)=x-8 ; g(x)=2x^2 What is the domain of f-g? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is x . (Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The domain is x x is any real number .$

Transcript text: For the given functions f and g , complete parts (a)-(h). For parts (a)-(d), also find the domain. \[ f(x)=x-8 ; g(x)=2 x^{2} \] $\qquad$ What is the domain of $\mathrm{f}-\mathrm{g}$ ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is $\{x \mid$ $\square$ \}. (Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The domain is $\{x \mid x$ is any real number $\}$.

Solution

Solution Steps

To find the domain of $ f - g $, we need to determine the set of all $ x $ values for which both $ f(x) $ and $ g(x) $ are defined. Since $ f(x) = x - 8 $ and $ g(x) = 2x^2 $ are both polynomials, they are defined for all real numbers. Therefore, the domain of $ f - g $ is all real numbers.

Solution Approach

The domain of $ f - g $ is all real numbers because both $ f(x) $ and $ g(x) $ are defined for all real numbers.

Step 1: Define the Functions

We have the functions defined as follows: \[ f(x) = x - 8 \] \[ g(x) = 2x^2 \]

Step 2: Determine the Domain of $ f $ and $ g $

Both $ f(x) $ and $ g(x) $ are polynomial functions. Polynomial functions are defined for all real numbers. Therefore, the domain of each function is: \[ \text{Domain of } f: \{ x \mid x \in \mathbb{R} \} \] \[ \text{Domain of } g: \{ x \mid x \in \mathbb{R} \} \]

Step 3: Find the Domain of $ f - g $

To find the domain of $ f - g $, we need to consider the intersection of the domains of $ f $ and $ g $. Since both functions are defined for all real numbers, the domain of $ f - g $ is also: \[ \text{Domain of } (f - g): \{ x \mid x \in \mathbb{R} \} \]