The functions \( f \) and \( g \) are defined as follows. \[ \begin{array}{l} f(x)=\frac{x^{2}}{x+7} \\ g(x)=\frac{x+9}{x^{2}-81} \end{array} \] For each function, find the domain. Write each answer as an interval or union of intervals. Domain of \( f \) : Domain of \( g \)

To find the domain of \( f(x) = \frac{x^{2}}{x+7} \), we need to determine where the function is defined. This rational function is undefined where the denominator is zero. Setting the denominator equal to zero gives us: \[ x + 7 = 0 \implies x = -7. \] Thus, the domain of \( f \) is all real numbers except \( -7 \): \[ \text{Domain of } f: (-\infty, -7) \cup (-7, \infty). \] Next, for \( g(x) = \frac{x+9}{x^{2}-81} \), we again look for values that make the denominator zero. The expression \( x^2 - 81 \) factors as \( (x-9)(x+9) \), so we set it to zero: \[ x^2 - 81 = 0 \implies x = 9 \text{ or } x = -9. \] Therefore, the function \( g \) is undefined at \( x = 9 \) and \( x = -9 \). The domain of \( g \) excludes these points: \[ \text{Domain of } g: (-\infty, -9) \cup (-9, 9) \cup (9, \infty). \]