If \( f(x)=\frac{7}{x+5} \) and \( g(x)=\frac{x+75}{x^{2}-25} \), determine \( (f+g)(x) \) \( \begin{array}{l}(f+g)(x)=\frac{x+82}{x^{2}-25} \\ (f+g)(x)=\frac{x+82}{x+5} \\ (f+g)(x)=\frac{8}{x^{2}-25} \\ (f+g)(x)=\frac{8}{x-5}\end{array} \)

To find \( (f + g)(x) \), we need to combine the functions \( f(x) \) and \( g(x) \). First, let's express both functions clearly: - \( f(x) = \frac{7}{x + 5} \) - \( g(x) = \frac{x + 75}{x^2 - 25} \) We will find a common denominator to add them together. The denominator for \( g(x) \) can be factored as \( x^2 - 25 = (x - 5)(x + 5) \). Now we can write \( (f + g)(x) \): \[ (f + g)(x) = f(x) + g(x) = \frac{7}{x + 5} + \frac{x + 75}{(x - 5)(x + 5)} \] Next, we need to express \( f(x) \) with the common denominator: \[ f(x) = \frac{7(x - 5)}{(x - 5)(x + 5)} = \frac{7x - 35}{(x - 5)(x + 5)} \] Now we can add: \[ (f + g)(x) = \frac{7x - 35 + x + 75}{(x - 5)(x + 5)} = \frac{8x + 40}{(x - 5)(x + 5)} \] Finally, we can factor out a common term from the numerator: \[ (f + g)(x) = \frac{8(x + 5)}{(x - 5)(x + 5)} \] Now, cancelling \( x + 5 \) (provided \( x \neq -5 \)): \[ (f + g)(x) = \frac{8}{x - 5}, \text{ for } x \neq -5 \] So, the answer is: \[ (f + g)(x) = \frac{8}{x - 5} \]