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).