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

To find the domain of the function \( h(x) = (f \times g)(x) \), where:

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

we need to determine the domain of both \( f(x) \) and \( g(x) \) and then find the intersection of those domains.

### Step 1: Find the domain of \( f(x) = \sqrt{x} \)
For the square root function \( \sqrt{x} \), the argument inside the square root must be non-negative. This gives the condition:

\[
x \geq 0.
\]

Thus, the domain of \( f(x) = \sqrt{x} \) is:
\[
\text{Domain of } f(x) = [0, \infty).
\]

### Step 2: Find the 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: Find the domain of \( h(x) = (f \times g)(x) \)
The function \( h(x) = (f \times g)(x) = \sqrt{x}(x^2 - 64) \) will only be defined where both \( f(x) \) and \( g(x) \) are defined. Since the domain of \( f(x) \) is \( [0, \infty) \) and the domain of \( g(x) \) is \( (-\infty, \infty) \), the domain of \( h(x) \) is the intersection of these two domains.

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

\[
[0, \infty).
\]

### Final Answer in Interval Notation:
\[
\boxed{[0, \infty)}
\]

Would you like further details or clarification?

---

Here are 5 related questions:

1. What is the range of \( f(x) = \sqrt{x} \)?
2. How does the behavior of \( g(x) = x^2 - 64 \) affect the graph of \( h(x) \)?
3. What is the domain of \( h(x) = \frac{f(x)}{g(x)} \)?
4. How would the domain change if \( f(x) = \sqrt{x + 1} \) instead?
5. What are the intercepts of \( h(x) \) based on the given functions?

**Tip**: Always check for restrictions such as square roots and denominators when determining a function's domain.

This problem guides you through finding the domain of h(x) = sqrt(x) * (x^2 - 64). It explains key concepts such as square root and polynomial functions, the domain of a function, and how to find the intersection of domains. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation