Questions: For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain. f(x)=3x+8; g(x)=7x-5 (a) Find (f+g)(x). (f+g)(x)=10x+3 (Simplify your answer.) 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 . (b) Find (f-g)(x) (f-g)(x)= (Simplify your answer.) 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)=3x+8; g(x)=7x-5 (a) Find (f+g)(x). (f+g)(x)=10x+3 (Simplify your answer.) 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 . (b) Find (f-g)(x) (f-g)(x)= (Simplify your answer.) 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. \[ f(x)=3 x+8 ; g(x)=7 x-5 \] (a) Find $(f+g)(x)$. \[ (f+g)(x)=10 x+3 \text { (Simplify your answer.) } \] 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 $\{x \mid \quad\}$ (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 $\}$. (b) Find $(f-g)(x)$ $(\mathrm{f}-\mathrm{g})(\mathrm{x})=$ $\square$ (Simplify your answer.) 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 $\{x \mid$ $\square$ (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 $\}
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Solution

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Solution Steps

Solution Approach

To solve the given problems, we need to perform basic operations on the functions \( f(x) = 3x + 8 \) and \( g(x) = 7x - 5 \).

(a) To find \((f+g)(x)\), we add the two functions together. The domain of the sum of two linear functions is all real numbers, as there are no restrictions like division by zero or square roots of negative numbers.

(b) To find \((f-g)(x)\), we subtract \( g(x) \) from \( f(x) \). Similarly, the domain of the difference of two linear functions is all real numbers.

Step 1: Find \((f+g)(x)\)

To find \((f+g)(x)\), we add the two functions: \[ f(x) + g(x) = (3x + 8) + (7x - 5) = 10x + 3 \]

Step 2: Determine the Domain of \(f+g\)

The domain of the sum of two linear functions is all real numbers. Therefore, we can express the domain as: \[ \{x \mid x \text{ is any real number}\} \]

Step 3: Find \((f-g)(x)\)

To find \((f-g)(x)\), we subtract \(g(x)\) from \(f(x)\): \[ f(x) - g(x) = (3x + 8) - (7x - 5) = 3x + 8 - 7x + 5 = 13 - 4x \]

Step 4: Determine the Domain of \(f-g\)

Similar to the sum, the domain of the difference of two linear functions is also all real numbers. Thus, we express the domain as: \[ \{x \mid x \text{ is any real number}\} \]

Final Answer

For part (a), \((f+g)(x) = 10x + 3\) and the domain is \(\{x \mid x \text{ is any real number}\}\).
For part (b), \((f-g)(x) = 13 - 4x\) and the domain is \(\{x \mid x \text{ is any real number}\}\).

\[ \boxed{(f+g)(x) = 10x + 3, \quad (f-g)(x) = 13 - 4x} \]

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