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The functions \( f \) and \( g \) are defined as follows. \[ f(x)=\frac{x^{2}}{x+8} \] \( g(x)=\frac{x+6}{x^{2}-36} \) For each function, find the domain. Write each answer as an interval or union of intervals. Domain of \( f: \) Domain of \( g: \square \)

Ask by Cook Gordon. in the United States
Mar 18,2025

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Domain of \( f: (-\infty,-8) \cup (-8,\infty) \) Domain of \( g: (-\infty,-6) \cup (-6,6) \cup (6,\infty) \)

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Beyond the Answer

To find the domain of the function \( f(x) = \frac{x^{2}}{x+8} \), we need to ensure that the denominator does not equal zero. Thus, we solve: \[ x + 8 \neq 0 \Rightarrow x \neq -8 \] Since there are no other restrictions (the numerator can take any value), the domain of \( f \) is all real numbers except \( -8 \): Domain of \( f: (-\infty, -8) \cup (-8, \infty) \) Now, for the function \( g(x) = \frac{x+6}{x^{2}-36} \), we need to look for points where the denominator is zero. We set the denominator to zero and solve: \[ x^{2} - 36 = 0 \Rightarrow (x-6)(x+6) = 0 \Rightarrow x = 6 \text{ or } x = -6 \] Therefore, we exclude these values from the domain: Domain of \( g: (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \)

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