Give the domain for  
p
(
x
)
=
x
+
9
x
2
−
9
x
+
18
  in interval notation.

To determine the domain of the function \( p(x) = \frac{x+9}{x^2 - 9x + 18} \), we must find the values of \( x \) that make the denominator equal to zero, as division by zero is undefined.

The denominator is \( x^2 - 9x + 18 \). To find when this equals zero, we solve the quadratic equation:

\[
x^2 - 9x + 18 = 0
\]

We use the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(18)}}{2(1)}
\]

Simplifying:

\[
x = \frac{9 \pm \sqrt{81 - 72}}{2}
\]
\[
x = \frac{9 \pm \sqrt{9}}{2}
\]
\[
x = \frac{9 \pm 3}{2}
\]

Thus, the two solutions are:

\[
x = \frac{9 + 3}{2} = 6 \quad \text{and} \quad x = \frac{9 - 3}{2} = 3
\]

Therefore, the function is undefined at \( x = 3 \) and \( x = 6 \).

### Domain
The domain is all real numbers except 3 and 6. In interval notation, the domain is:

\[
(-\infty, 3) \cup (3, 6) \cup (6, \infty)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve a quadratic equation using the quadratic formula?
2. What does it mean for a function to be undefined?
3. Can you give examples of other rational functions and their domains?
4. How can you check if your solutions to a quadratic equation are correct?
5. What is interval notation, and how is it used in different contexts?

**Tip:** Always check the denominator of a rational function to identify possible restrictions in its domain.

Give the domain for p(x) = \frac{x+9}{x^2 - 9x + 18} in interval notation.

Learn how to find the domain of the rational function p(x) = (x+9)/(x^2 - 9x + 18) by solving for restrictions in the denominator. This problem uses key algebraic concepts like solving quadratic equations with the quadratic formula and finding the domain in interval notation. Suitable for students in Grades 9-12.

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Give the domain for
p ( x )

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