The domain and range of a function are fundamental concepts in mathematics, particularly in algebra and calculus. They describe the set of possible inputs and outputs for a function.

1. Domain

The domain of a function is the set of all possible input values (often represented by $x$) for which the function is defined.

• For a function $f(x)$, you look for all $x$-values that do not cause any mathematical issues in the function (like division by zero or taking the square root of a negative number if we’re dealing with real numbers).
• For example, the domain of $f(x) = \frac{1}{x}$ is all real numbers except $x = 0$, since dividing by zero is undefined.

2. Range

The range of a function is the set of all possible output values (often represented by $y$) that the function can produce.

• For the same function $f(x) = \frac{1}{x}$, the range is all real numbers except $y = 0$, because $f(x)$ can never equal zero.

In summary:

• Domain: The set of possible $x$-values (inputs).
• Range: The set of possible $y$-values (outputs).

Would you like more details on how to find the domain and range for specific types of functions?

Here are 5 questions that build on this explanation:

1. How do you determine the domain of a polynomial function?
2. What is the range of a quadratic function, and how can it be calculated?
3. How does the domain and range change when a function is transformed (e.g., shifted or stretched)?
4. What are the domain and range for trigonometric functions like sine and cosine?
5. How does understanding domain and range help in solving real-world problems?

Tip: For functions with square roots, always ensure the expression inside the root is non-negative to find the domain in the real number system.