Final answer:
A relation is a function if every input is associated with exactly one output. This can be visually tested using the vertical line test on a graph or by ensuring that no two different ordered pairs in a set have the same first element when looking at mathematical expressions.
Step-by-step explanation:
To identify if a relation is a function or not, one must understand the definition of a function. A function is a specific type of relation where every input is associated with exactly one output. This means that for each value in the domain, there is one and only one corresponding value in the codomain. The classic test for a function is the vertical line test when analyzing a graph.
If you can draw a vertical line anywhere on the graph and it intersects the graph in at most one point, then the relation represented by the graph is a function. Mathematically, you can look at a set of ordered pairs; if no two different ordered pairs have the same first element, then the set of ordered pairs represents a function. Similarly, when you have an equation, you can solve for the output (often represented as 'y') for different inputs (represented as 'x') to verify if each input gives a single output.
For example, the equation of a line such as y = mx + b where 'm' is the slope and 'b' is the y-intercept always represents a function, because for each value of 'x', there is exactly one corresponding value of 'y'. However, an equation like y^2 = x does not represent a function because for some values of 'x', there can be two different values for 'y' (both positive and negative square roots).