given f(x)=squareroot of x and g(x)=x squared minus 64, find the domain of h(x)=(f times g)(x). write in interval notation

To find the domain of \( h(x) = (f \times g)(x) \), we need to determine the domains of \( f(x) \) and \( g(x) \) individually, then find the intersection of those domains because the product \( h(x) = f(x) \cdot g(x) \) will only be valid where both functions are defined.

### Step 1: Domain of \( f(x) = \sqrt{x} \)
For \( f(x) \), the square root function is only defined when \( x \geq 0 \) because the square root of a negative number is not a real number. Therefore, the domain of \( f(x) \) is:
\[
D_f = [0, \infty)
\]

### Step 2: Domain of \( g(x) = x^2 - 64 \)
The function \( g(x) = x^2 - 64 \) is a quadratic function, and quadratic functions are defined for all real values of \( x \). Thus, the domain of \( g(x) \) is:
\[
D_g = (-\infty, \infty)
\]

### Step 3: Domain of \( h(x) = f(x) \cdot g(x) \)
The function \( h(x) = f(x) \cdot g(x) \) will only be defined where both \( f(x) \) and \( g(x) \) are defined. Since \( f(x) \) restricts the domain to \( x \geq 0 \), the domain of \( h(x) \) is the intersection of \( D_f \) and \( D_g \), which is:
\[
D_h = [0, \infty)
\]

### Final Answer (in interval notation):
The domain of \( h(x) \) is:
\[
[0, \infty)
\]

Do you want more details or have any questions? Here are five related questions:

1. How would the domain change if \( f(x) = \sqrt{x-1} \) instead?
2. What is the range of \( g(x) = x^2 - 64 \)?
3. How would you graph \( h(x) = f(x) \cdot g(x) \)?
4. How does the product of two functions affect the domain of the resulting function?
5. What happens to the domain if \( g(x) = \frac{1}{x^2 - 64} \)?

**Tip:** When combining functions, always check the individual domains to ensure the result is well-defined!

This guide explains how to find the domain of h(x) = (f * g)(x), where f(x) is the square root of x and g(x) is x squared minus 64. Learn the key concepts of function domains and quadratic functions in this step-by-step solution. Suitable for high school students.

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