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se the given functions to find the requested information. \( f(x)=-3 x+7 \) and \( g(x)=\frac{2}{x-1} \). Find \( (f+g)(x) \) and the domain of \( f+g \).

Ask by Daniels Lang. in the United States
Mar 16,2025

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\[ (f+g)(x) = -3x + 7 + \frac{2}{x-1}, \quad \text{and the domain is all real numbers except } x = 1. \]

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Bonus Knowledge

To find \( (f+g)(x) \), we simply add the two functions together: \[ (f+g)(x) = f(x) + g(x) = (-3x + 7) + \left(\frac{2}{x-1}\right) \] This simplifies to: \[ (f+g)(x) = -3x + 7 + \frac{2}{x-1} \] Now, to determine the domain of \( f+g \), we need to identify where our resulting function is defined. Since \( g(x) \) includes a fraction, it's important to exclude any value that makes the denominator zero. The expression \( x-1 \neq 0 \) implies \( x \neq 1 \). Therefore, the domain of \( f+g \) is all real numbers except \( x=1 \): \[ \text{Domain: } (-\infty, 1) \cup (1, \infty) \]

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