Can you give me a sample of domain and range

### Sample of Domain and Range

**Example Function:**

Let's consider the function: 

\[
f(x) = \sqrt{x - 2}
\]

**Domain:**

The domain of a function is the set of all possible input values (x-values) that the function can accept.

For \( f(x) = \sqrt{x - 2} \), the expression inside the square root must be non-negative because the square root of a negative number is not a real number. Thus, we have:

\[
x - 2 \geq 0 \implies x \geq 2
\]

**Domain:** \(\{ x \in \mathbb{R} \mid x \geq 2 \}\) or in interval notation: \([2, \infty)\).

**Range:**

The range of a function is the set of all possible output values (y-values) that the function can produce.

Since \( f(x) = \sqrt{x - 2} \), as \( x \) increases starting from 2, the square root produces all non-negative numbers. The smallest value is 0 when \( x = 2 \), and it increases without bound as \( x \) increases.

**Range:** \(\{ y \in \mathbb{R} \mid y \geq 0 \}\) or in interval notation: \([0, \infty)\).

### Summary:
- **Domain:** \([2, \infty)\)
- **Range:** \([0, \infty)\)

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Would you like to see more examples or have specific questions? Here are some related questions:

1. What is the domain and range of \( f(x) = \frac{1}{x-1} \)?
2. How do you find the domain and range of a quadratic function?
3. Can you explain domain and range for a trigonometric function like \( \sin(x) \)?
4. How does the presence of a square root affect the domain and range?
5. What are domain and range restrictions in real-world contexts?

**Tip:** Always check the type of expression inside the function to determine restrictions on the domain, like square roots (must be non-negative) or fractions (denominator can't be zero).

Can you give me a sample of domain and range?

Explore the domain and range of the function f(x) = √(x - 2). This example demonstrates how to determine the set of possible input values (domain) and output values (range) for functions involving square roots. Suitable for high school students in Grades 9-12.

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