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

We are given two functions:
- \( f(x) = \sqrt{x} \)
- \( g(x) = x^2 - 64 \)

We need to find the domain of the composite function \( h(x) = (f \times g)(x) \), which means:
\[
h(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x^2 - 64)
\]

### Step 1: Domain of \( f(x) \)
The function \( f(x) = \sqrt{x} \) is defined only for non-negative values of \( x \). Therefore, the domain of \( f(x) \) is:
\[
x \geq 0
\]
This means the domain of \( f(x) \) is \( [0, \infty) \).

### Step 2: Domain of \( g(x) \)
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:
\[
(-\infty, \infty)
\]

### Step 3: Domain of \( h(x) \)
Since \( h(x) = f(x) \cdot g(x) \), the domain of \( h(x) \) must satisfy the domain restrictions of both \( f(x) \) and \( g(x) \). Specifically, \( x \) must satisfy:
\[
x \geq 0 \quad (\text{from the domain of } f(x))
\]

Thus, the domain of \( h(x) \) is the intersection of the domains of \( f(x) \) and \( g(x) \), which is:
\[
[0, \infty)
\]

### Final Answer:
The domain of \( h(x) = \sqrt{x}(x^2 - 64) \) in interval notation is:
\[
[0, \infty)
\]

Would you like to go over any specific part of the explanation?

Here are 5 related questions you can explore:
1. How do you find the domain of a composite function involving division?
2. What happens if \( f(x) \) is a rational function instead of a square root function?
3. How do you find the range of the function \( h(x) \)?
4. Can you graph \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 64 \)?
5. How would the domain change if \( g(x) \) had a square root?

**Tip:** Always check for any restrictions from each individual function when finding the domain of a composite function.

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

Explore how to find the domain of the composite function h(x) = sqrt(x) * (x^2 - 64) with a thorough step-by-step explanation. Learn about the domain of square root functions and polynomial functions, suitable for high school students in Grades 10-12.

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