Questions: The functions (f) and (g) are defined as follows. [ f(x)=fracx+7x^2-49 ] [ g(x)=fracxx^2+9 ] For each function, find the domain. Write each answer as an interval or union of intervals. Domain of (f) : (square) Domain of (g) : (square)

The functions (f) and (g) are defined as follows.

[
f(x)=fracx+7x^2-49
]
[
g(x)=fracxx^2+9
]

For each function, find the domain.
Write each answer as an interval or union of intervals.

Domain of (f) : (square)

Domain of (g) : (square)
Transcript text: The functions $f$ and $g$ are defined as follows. \[ \begin{array}{l} f(x)=\frac{x+7}{x^{2}-49} \\ g(x)=\frac{x}{x^{2}+9} \end{array} \] For each function, find the domain. Write each answer as an interval or union of intervals. Domain of $f$ : $\square$ Domain of $g$ : $\square$
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Solution

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Solution Steps

To find the domain of a function, we need to determine the values of \( x \) for which the function is defined. For rational functions, this means identifying the values of \( x \) that do not make the denominator zero.

  1. Domain of \( f(x) \): The function \( f(x) = \frac{x+7}{x^2-49} \) has a denominator of \( x^2 - 49 \). Set the denominator equal to zero and solve for \( x \) to find the values that are not in the domain.

  2. Domain of \( g(x) \): The function \( g(x) = \frac{x}{x^2+9} \) has a denominator of \( x^2 + 9 \). Set the denominator equal to zero and solve for \( x \) to find the values that are not in the domain.

Step 1: Find the Domain of \( f(x) \)

The function \( f(x) = \frac{x+7}{x^2-49} \) has a denominator of \( x^2 - 49 \). To find the values that make the denominator zero, we solve the equation:

\[ x^2 - 49 = 0 \]

This factors to:

\[ (x - 7)(x + 7) = 0 \]

Thus, the solutions are \( x = 7 \) and \( x = -7 \). Therefore, the domain of \( f(x) \) excludes these values, which can be expressed as:

\[ \text{Domain of } f: (-\infty, -7) \cup (-7, 7) \cup (7, \infty) \]

Step 2: Find the Domain of \( g(x) \)

The function \( g(x) = \frac{x}{x^2+9} \) has a denominator of \( x^2 + 9 \). To find the values that make the denominator zero, we solve the equation:

\[ x^2 + 9 = 0 \]

This yields complex solutions:

\[ x = \pm 3i \]

Since there are no real solutions, the domain of \( g(x) \) includes all real numbers:

\[ \text{Domain of } g: (-\infty, \infty) \]

Final Answer

The domains of the functions are:

  • Domain of \( f \): \((- \infty, -7) \cup (-7, 7) \cup (7, \infty)\)
  • Domain of \( g \): \((- \infty, \infty)\)

Thus, the final answer is:

\[ \boxed{\text{Domain of } f: (-\infty, -7) \cup (-7, 7) \cup (7, \infty), \text{ Domain of } g: (-\infty, \infty)} \]

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