In mathematics, **domain** and **range** are fundamental concepts used to describe the behavior of functions.

### 1. Domain
The **domain** of a function is the set of all possible input values (independent variable) for which the function is defined. In simpler terms, it refers to all the values of \( x \) for which the function \( f(x) \) produces a real output.

For example:
- For the function \( f(x) = \frac{1}{x} \), the domain is all real numbers except \( x = 0 \), because dividing by zero is undefined.
- For \( f(x) = \sqrt{x} \), the domain is \( x \geq 0 \), since the square root of a negative number is not a real number.

### 2. Range
The **range** of a function is the set of all possible output values (dependent variable) that the function can produce. In other words, it is all the values that \( f(x) \) can take as \( x \) varies over the domain.

For example:
- For \( f(x) = x^2 \), the range is \( y \geq 0 \), because squaring any real number produces a non-negative result.
- For \( f(x) = \sin(x) \), the range is \( -1 \leq y \leq 1 \), as the sine function only outputs values within this interval.

Would you like more examples or details on any specific type of function?

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**Related Questions:**
1. What is the domain of \( f(x) = \frac{1}{x-2} \)?
2. How do you find the range of a quadratic function?
3. Can a function have an infinite range?
4. What is the domain of a logarithmic function like \( \log(x) \)?
5. How does the domain of trigonometric functions like \( \tan(x) \) differ from others?

**Tip:** Always check for restrictions like division by zero or square roots of negative numbers when determining the domain of a function.

Explore the concepts of domain and range in functions, with examples such as quadratic, trigonometric, and rational functions. Understand how to determine the domain and range of various types of functions, and learn tips for handling restrictions. Ideal for students in Grades 9-12.

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