The functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{x^{2}}{x+4} \] \[ g(x)=\frac{x-8}{x^{2}-64} \] For each function, find the domain. Write each answer as an interval or union of intervals.

To find the domain of the functions \( f(x) \) and \( g(x) \), we need to identify the values of \( x \) for which each function is defined. ### Domain of \( f(x) \) The function \( f(x) = \frac{x^2}{x + 4} \) is a rational function. The domain of a rational function is all real numbers except where the denominator is zero. 1. Set the denominator equal to zero: \[ x + 4 = 0 \] Solving for \( x \): \[ x = -4 \] Thus, the function \( f(x) \) is undefined at \( x = -4 \). Therefore, the domain of \( f(x) \) is all real numbers except \( -4 \). In interval notation, the domain of \( f(x) \) is: \[ (-\infty, -4) \cup (-4, \infty) \] ### Domain of \( g(x) \) The function \( g(x) = \frac{x - 8}{x^2 - 64} \) is also a rational function. We need to find where the denominator is zero. 1. Set the denominator equal to zero: \[ x^2 - 64 = 0 \] This can be factored as: \[ (x - 8)(x + 8) = 0 \] Solving for \( x \): \[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \] \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \] Thus, the function \( g(x) \) is undefined at \( x = 8 \) and \( x = -8 \). Therefore, the domain of \( g(x) \) is all real numbers except \( -8 \) and \( 8 \). In interval notation, the domain of \( g(x) \) is: \[ (-\infty, -8) \cup (-8, 8) \cup (8, \infty) \] ### Summary of Domains - The domain of \( f(x) \) is: \[ (-\infty, -4) \cup (-4, \infty) \] - The domain of \( g(x) \) is: \[ (-\infty, -8) \cup (-8, 8) \cup (8, \infty) \]