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?

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.