The functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{x}{x^{2}+16} \] \[ g(x)=\frac{x-3}{x^{2}+3 x-18} \] For each function, find the domain. Write each answer as an interval or union of intervals. Domain of \( f: \square \) Domain of \( g \) :

To find the domains of the functions \( f(x) \) and \( g(x) \), we need to identify the values of \( x \) that do not make the functions undefined. For the function \( f(x) = \frac{x}{x^2 + 16} \), the denominator \( x^2 + 16 \) is always positive since the minimum value of \( x^2 \) is 0 and thus \( x^2 + 16 \geq 16 \). Therefore, there are no restrictions on \( x \). The domain of \( f \) is all real numbers. Domain of \( f: (-\infty, \infty) \) For the function \( g(x) = \frac{x - 3}{x^2 + 3x - 18} \), we need to find when the denominator \( x^2 + 3x - 18 \) is not equal to zero. We can factor this quadratic: \[ x^2 + 3x - 18 = (x - 3)(x + 6) \] Setting this equal to zero gives us the roots \( x = 3 \) and \( x = -6 \), so the function is undefined at these values. Thus, the domain of \( g \) excludes these points. Domain of \( g: (-\infty, -6) \cup (-6, 3) \cup (3, \infty) \)