The functions $$ f $$ and $$ g $$ are defined as follows. $$ f(x)=\frac{x+9}{x^{2}+17 x+72} $$ $$ g(x)=\frac{x^{2}}{x-3} $$ For each function, find the domain. Write each answer as an interval or union of intervals.

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- Domain of $$ f(x) $$: $$ (-\infty, -9) \cup (-9, -8) \cup (-8, \infty) $$ - Domain of $$ g(x) $$: $$ (-\infty, 3) \cup (3, \infty) $$ 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 $$ f(x) $$ is given by: $$ f(x) = \frac{x + 9}{x^2 + 17x + 72} $$ The function is defined as long as the denominator is not zero. Therefore, we need to find the values of $$ x $$ that make the denominator zero: $$ x^2 + 17x + 72 = 0 $$ To solve this quadratic equation, we can factor it: $$ x^2 + 17x + 72 = (x + 8)(x + 9) = 0 $$ Setting each factor to zero gives us: $$ x + 8 = 0 \quad \Rightarrow \quad x = -8 $$ $$ x + 9 = 0 \quad \Rightarrow \quad x = -9 $$ Thus, the function $$ f(x) $$ is undefined at $$ x = -8 $$ and $$ x = -9 $$. The domain of $$ f(x) $$ is all real numbers except $$ -8 $$ and $$ -9 $$. In interval notation, this is: $$ (-\infty, -9) \cup (-9, -8) \cup (-8, \infty) $$ ### Domain of $$ g(x) $$ The function $$ g(x) $$ is given by: $$ g(x) = \frac{x^2}{x - 3} $$ This function is defined as long as the denominator is not zero. Therefore, we need to find the values of $$ x $$ that make the denominator zero: $$ x - 3 = 0 \quad \Rightarrow \quad x = 3 $$ Thus, the function $$ g(x) $$ is undefined at $$ x = 3 $$. The domain of $$ g(x) $$ is all real numbers except $$ 3 $$. In interval notation, this is: $$ (-\infty, 3) \cup (3, \infty) $$ ### Summary of Domains - The domain of $$ f(x) $$ is: $$ (-\infty, -9) \cup (-9, -8) \cup (-8, \infty) $$ - The domain of $$ g(x) $$ is: $$ (-\infty, 3) \cup (3, \infty) $$

The functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{x+9}{x^{2}+17 x+72} \] \[ g(x)=\frac{x^{2}}{x-3} \] For each function, find the domain. Write each answer as an interval or union of intervals.

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The functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{x+9}{x^{2}+17 x+72} \] \[ g(x)=\frac{x^{2}}{x-3} \] For each function, find the domain. Write each answer as an interval or union of intervals.

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