https://file.aiquickdraw.com/mathbot/file/17267659465902q6b6bs6.jpg

The function \( f(x) \) given in the image is defined as:
\[ f(x) = \frac{x+7}{x^2-81} \]

To find the values of \( x \) that are NOT in the domain of \( f \), we need to look for values that would make the denominator zero, as division by zero is undefined.

The denominator is \( x^2 - 81 \). We set this equal to zero and solve for \( x \):
\[ x^2 - 81 = 0 \]
\[ x^2 = 81 \]
\[ x = \pm \sqrt{81} \]
\[ x = \pm 9 \]

Therefore, the values of \( x \) that are NOT in the domain of \( f \) are \( x = 9 \) and \( x = -9 \). These are the values where the function \( f(x) \) would be undefined.

Do you have any questions about this explanation, or is there anything else you'd like to know? Here are some further questions you might consider:

1. What happens to the value of \( f(x) \) as \( x \) approaches values not in its domain?
2. How does the graph of \( f(x) \) look near the points where \( x = 9 \) and \( x = -9 \)?
3. Can you calculate the limit of \( f(x) \) as \( x \) approaches 9 and -9?
4. What are the asymptotes of the graph of \( f(x) \)?
5. How would the domain change if the function were instead \( f(x) = \frac{x+7}{x^2 + 81} \)?

**Tip**: To visually verify your understanding of a function's domain, consider sketching its graph, identifying regions where the function does not exist.

Find all values of x that are NOT in the domain of the function f(x) = (x + 7) / (x^2 - 81).

Discover how to find the values of x that are not in the domain of the rational function f(x) = (x + 7) / (x^2 - 81). This problem illustrates key algebraic concepts such as domain restrictions and solving quadratic equations. Ideal for students in Grades 10-12.

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