h(x)= square root of x-3 domain and range

To find the domain and range of the function \( h(x) = \sqrt{x - 3} \), we need to consider the properties of the square root function and the conditions under which the expression inside the square root is defined.

### Domain:
The expression inside the square root, \( x - 3 \), must be non-negative because the square root of a negative number is not defined in the set of real numbers.

\[ x - 3 \geq 0 \]
\[ x \geq 3 \]

So, the domain of \( h(x) \) is all real numbers \( x \) such that \( x \geq 3 \).

### Domain in Interval Notation:
\[ [3, \infty) \]

### Range:
For the square root function \( \sqrt{x - 3} \), the output is always non-negative because the square root of any non-negative number is also non-negative.

- When \( x = 3 \), \( h(x) = \sqrt{3 - 3} = 0 \).
- As \( x \) increases, \( h(x) \) also increases because the square root function is increasing.

Therefore, the range of \( h(x) \) is all non-negative real numbers.

### Range in Interval Notation:
\[ [0, \infty) \]

### Summary:
- **Domain**: \( [3, \infty) \)
- **Range**: \( [0, \infty) \)

Would you like more details or have any questions? Here are some related questions:

1. How do you determine the domain of a function involving a square root?
2. What are the steps to find the range of a function?
3. How would the domain and range change if the function were \( h(x) = \sqrt{3 - x} \)?
4. Can you explain why the square root function only outputs non-negative values?
5. How do you graph the function \( h(x) = \sqrt{x - 3} \) and verify the domain and range?

**Tip:** When finding the domain of functions involving square roots, always set the expression inside the square root greater than or equal to zero and solve for the variable.

h(x) = square root of x - 3 domain and range

Explore how to determine the domain and range of the function h(x) = √(x - 3). This problem reviews the properties of square root functions, covering key algebraic concepts such as finding domain restrictions and non-negative ranges. Ideal for students in Grades 9-11.

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