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For the given functions \( f \) and \( g \), complete parts (a)-(h). For parts (a)-(d), also find the domain. \( f(x)=5 x+2 ; g(x)=7 x-8 \) (a) Find \( (f+g)(x) \). \[ (f+g)(x)=: 12 x-6 \text { (Simplify your answer.) } \] 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 \( \{x \mid \) (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 \( \} \).

Ask by Chavez Vaughn. in the United States
Feb 19,2025

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(a) \( (f+g)(x) = 12x - 6 \) Domain of \( f+g \): All real numbers.

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We are given   f(x) = 5x + 2 and g(x) = 7x – 8. Step 1. To find (f + g)(x), add the expressions:   (f + g)(x) = f(x) + g(x)     = (5x + 2) + (7x – 8)     = 5x + 7x + 2 – 8     = 12x – 6 Step 2. Domain of f + g: Both functions f and g are defined for all real numbers (since they are polynomials, which have no domain restrictions). Therefore, their sum is defined for all real x. So, the correct choice is: B. The domain is { x | x is any real number }. In summary:  (a) (f + g)(x) = 12x – 6 and its domain is all real numbers.

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Beyond the Answer

The domain of \( f+g \) is \( B. \) The domain is \( \{x \mid x \) is any real number \( \} \). This is because both functions \( f(x) \) and \( g(x) \) are linear functions, which means they are defined for all real numbers. Now, let’s consider the simplest way to visualize this! Imagine you're plotting the graphs of \( f(x) \) and \( g(x) \) on a coordinate plane. Both will be straight lines that stretch infinitely in both directions, obviously enjoying their fullness with no interruptions—just like your access to all real numbers!

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