h(x)= square root of x / (x^2+8x+15), find domain in interval notation

To find the domain of the function \( h(x) = \frac{\sqrt{x}}{x^2 + 8x + 15} \), we need to consider the following:

1. **Square root constraint**: The square root function \( \sqrt{x} \) is only defined for non-negative values of \( x \), so we must have:
   \[
   x \geq 0
   \]

2. **Denominator constraint**: The denominator \( x^2 + 8x + 15 \) cannot be zero, so we must find where this quadratic expression equals zero and exclude those values from the domain. Let's factor the quadratic:
   \[
   x^2 + 8x + 15 = (x + 3)(x + 5)
   \]
   So, \( x^2 + 8x + 15 = 0 \) when \( x = -3 \) or \( x = -5 \). These values are where the denominator is zero, but since \( x \geq 0 \), we don't need to worry about these negative values.

Thus, the domain of \( h(x) \) is:
\[
x \geq 0 \quad \text{and} \quad x \neq -3, -5
\]
Since these problematic values are negative and excluded by the square root constraint, the domain of \( h(x) \) is simply:
\[
[0, \infty)
\]

**Conclusion**: The domain of \( h(x) \) is \( [0, \infty) \).

Would you like more details or have any questions about this solution?

Here are 5 related questions to expand your understanding:
1. How do we factor different types of quadratics?
2. What are the restrictions for square root functions?
3. How do we solve inequalities involving square roots?
4. What other conditions can limit the domain of a function?
5. How does the denominator affect the domain of a rational function?

**Tip**: Always remember that square roots must have non-negative radicands, which often places limits on the domain of a function.

Explore how to find the domain of the function h(x) = sqrt(x) / (x^2 + 8x + 15). This problem involves understanding the domain restrictions of radical and rational functions. Step-by-step solution suitable for students in Grades 9-11.

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