given f(x)= squareroot of x and g(x)=xsquared minus 64, find the domain of h(x)=(f times g)(x). show your work and write answer in interval notation 

To find the domain of \( h(x) = (f \cdot g)(x) \), where

\[
f(x) = \sqrt{x} \quad \text{and} \quad g(x) = x^2 - 64,
\]

we need to consider the domains of both \( f(x) \) and \( g(x) \), and then find their intersection.

### Step 1: Domain of \( f(x) = \sqrt{x} \)
The square root function \( \sqrt{x} \) is defined only when the expression inside the square root is non-negative. Therefore, we must have:

\[
x \geq 0.
\]

Thus, the domain of \( f(x) \) is:

\[
\text{Domain of } f(x) = [0, \infty).
\]

### Step 2: Domain of \( g(x) = x^2 - 64 \)
The function \( g(x) = x^2 - 64 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of \( g(x) \) is:

\[
\text{Domain of } g(x) = (-\infty, \infty).
\]

### Step 3: Domain of \( h(x) = (f \cdot g)(x) \)
The function \( h(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x^2 - 64) \) is the product of \( f(x) \) and \( g(x) \). Since the product of two functions is only defined where both functions are defined, the domain of \( h(x) \) is the intersection of the domains of \( f(x) \) and \( g(x) \).

The domain of \( f(x) \) is \( [0, \infty) \), and the domain of \( g(x) \) is \( (-\infty, \infty) \). The intersection of these two domains is:

\[
\text{Domain of } h(x) = [0, \infty).
\]

Thus, the domain of \( h(x) \) is:

\[
[0, \infty)
\]

### Final Answer:
The domain of \( h(x) = (f \cdot g)(x) \) is \( [0, \infty) \).

---

Would you like further clarification or details on any part of the process? Here are some follow-up questions to consider:

1. What are the possible range values of \( h(x) \)?
2. How does multiplying two functions affect their overall domain?
3. What if \( f(x) \) was a different function? How would that change the domain of \( h(x) \)?
4. How does the domain of a square root function change with different radicands?
5. How would you graph \( h(x) \) and interpret its behavior over its domain?

**Tip:** Always remember to check for restrictions in each function before finding the domain of their combination!

Given f(x) = √x and g(x) = x² - 64, find the domain of h(x) = (f · g)(x). Show your work and write the answer in interval notation.

Discover how to find the domain of h(x) = √x · (x² - 64) by analyzing the square root function and polynomial. The solution walks through the steps of identifying the domain of both functions and finding their intersection, suitable for students in Grades 9-12.

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