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

Given the functions \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 64 \), we are tasked with finding the domain of \( h(x) = (f \cdot g)(x) \), where \( h(x) = f(x) \cdot g(x) \).

### Step-by-Step Solution:

1. **Determine the domain of \( f(x) \):**
   The function \( f(x) = \sqrt{x} \) requires that the argument under the square root is non-negative. Therefore, the domain of \( f(x) \) is:
   \[
   x \geq 0
   \]
   In interval notation, the domain of \( f(x) \) is:
   \[
   [0, \infty)
   \]

2. **Determine the domain of \( g(x) \):**
   The function \( g(x) = x^2 - 64 \) is a polynomial, which is defined for all real numbers. So, the domain of \( g(x) \) is:
   \[
   (-\infty, \infty)
   \]

3. **Determine the domain of \( h(x) = f(x) \cdot g(x) \):**
   Since \( h(x) = f(x) \cdot g(x) \), the domain of \( h(x) \) is restricted by the domain of \( f(x) \), because the product of two functions is only defined where both functions are defined. Therefore, the domain of \( h(x) \) is determined by the domain of \( f(x) \), which is \( [0, \infty) \).

### Conclusion:
The domain of \( h(x) = f(x) \cdot g(x) \) is:
\[
[0, \infty)
\]
This is because \( f(x) = \sqrt{x} \) restricts the domain to non-negative values of \( x \).

### Final Answer (Interval Notation):
\[
[0, \infty)
\]

Would you like more details or have any questions?

Here are 5 relative questions for further understanding:
1. What is the range of the function \( f(x) = \sqrt{x} \)?
2. How would the domain change if \( f(x) = \sqrt{x - 1} \)?
3. How does multiplying functions affect their domains in general?
4. Can the function \( g(x) = x^2 - 64 \) have any roots? If so, where?
5. How does the domain of \( f(x) \) influence composite functions?

**Tip**: Always check for restrictions like square roots or denominators when determining the domain of a function.

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

Solve the problem of finding the domain of the composite function h(x) = (f * g)(x), where f(x) = √x and g(x) = x² - 64. Learn the step-by-step process to determine the domain in interval notation, with a focus on the restrictions imposed by square roots and function multiplication. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation