Question

Domain vs range

Answer 1

In mathematics, the concepts of **domain** and **range** are fundamental to the understanding of functions.

### Domain
The **domain** of a function is the set of all possible input values (or independent variables) for which the function is defined. In other words, it includes all the x-values that you can use in the function without causing any inconsistencies, such as division by zero or taking the square root of a negative number (in the context of real numbers).

For example:
- For the function \( f(x) = \sqrt{x} \), the domain is \( x \geq 0 \) (since you cannot take the square root of a negative number).
- For the function \( g(x) = \frac{1}{x} \), the domain is \( x \neq 0 \) (since division by zero is undefined).

### Range
The **range** of a function is the set of all possible output values (or dependent variables) that the function can produce based on its domain. In other words, it includes all the y-values that correspond to the x-values in the domain when the function is evaluated.

For example:
- For the function \( f(x) = \sqrt{x} \), the range is \( y \geq 0 \) (since the square root of any non-negative number is non-negative).
- For the function \( g(x) = \frac{1}{x} \), the range is \( y \neq 0 \) (since the output can be any real number except zero).

### Summary
- **Domain:** Set of all possible input values.
- **Range:** Set of all possible output values.

Understanding the domain and range is crucial when analyzing functions, as it helps to determine the valid input and output values for that function.