In mathematics, particularly in the study of functions, the distinction between a function and a relation (or something that is not a function) is based on the relationship between inputs and outputs.
Function:
- Definition: A function is a specific type of relation that assigns exactly one output to each input from a given set (domain).
- Notation: Functions are often denoted as \( f(x) \), where \( f \) is the function name and \( x \) is the input.
- Uniqueness: For every element \( x \) in the domain, there is one and only one corresponding element \( y \) in the range. In other words, if \( f(a) = b \), then there cannot be another \( c \) such that \( f(a) = c \) for the same \( a \).
- Examples:
- \( f(x) = x^2 \) is a function because for each input \( x \), there is one output \( x^2 \).
- The relation \( g(x) = 2x + 3 \) is also a function since each input gives a single output.
Not a Function:
- Definition: A relation that does not assign exactly one output for every input is not a function.
- Multiple Outputs: If any input has more than one output, then it is not a function.
- Examples:
- The relation \((1, 2), (1, 3)\) fails to be a function because the input \( 1 \) corresponds to two different outputs: \( 2 \) and \( 3 \).
- The equation \( y^2 = x \) is not a function because for certain values of \( x \) (like \( 1 \)), there are two possible \( y \) values (both \( -1 \) and \( 1 \)).
Vertical Line Test:
A common graphical way to determine if a relation is a function is to use the Vertical Line Test. If any vertical line drawn on the graph of the relation intersects the graph at more than one point, then the relation is not a function.
In summary, the key difference is that a function must have a unique output for each input, while a relation that allows multiple outputs for a single input is not a function.