Find the domain of this function: f(x)= 1/ x 2 +x−72

To find the domain of the function:

\[
f(x) = \frac{1}{x^2 + x - 72}
\]

The domain consists of all values of \( x \) for which the function is defined. Since the function has a denominator, we need to ensure that the denominator is not zero. So, we set the denominator equal to zero and solve for \( x \):

\[
x^2 + x - 72 = 0
\]

Now solve the quadratic equation by factoring.

We need two numbers that multiply to \(-72\) and add up to \(1\). These numbers are \(9\) and \(-8\). Therefore, the factorization is:

\[
(x + 9)(x - 8) = 0
\]

Set each factor equal to zero:

\[
x + 9 = 0 \quad \Rightarrow \quad x = -9
\]
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]

So, the denominator is zero at \(x = -9\) and \(x = 8\). The function is undefined at these points.

Thus, the domain of the function is all real numbers except \(x = -9\) and \(x = 8\).

In interval notation, the domain is:

\[
(-\infty, -9) \cup (-9, 8) \cup (8, \infty)
\]

Would you like any additional details or have any further questions?

Here are 5 related questions:
1. How do you find the range of a rational function?
2. What happens to the function near the points \(x = -9\) and \(x = 8\)?
3. How do you factor quadratic expressions more efficiently?
4. What are vertical asymptotes, and how do they relate to domain restrictions?
5. Can you graph this function to visualize its behavior?

**Tip:** Always check the denominator of a rational function to find potential restrictions in the domain!

Find the domain of this function: f(x)= 1/ x^2 + x − 72

Discover how to find the domain of the rational function f(x) = 1/(x^2 + x - 72) using algebraic techniques such as factoring. Learn how to handle domain restrictions and solve quadratic equations. This solution is perfect for high school students in Grades 9-12.

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