Short Answer 
A linear function shows a consistent relationship between two variables with distinct outputs for each input, represented as a straight line on a graph. In contrast, a non-linear function may produce different outputs for the same input, indicating inconsistency. To identify linearity, observe if changes in input result in predictable changes in output, looking for straight-line patterns in graphs.
Step 1: Understanding Linear Functions
A linear function represents a relationship between two variables where there is a constant ratio between them, depicted as a straight line on a graph. For a function to be linear, it should not produce the same output (Y value) for the same input (X value). This means that for each unique input, there should be a distinct output, illustrating a proportional relationship.
Step 2: Identifying Non-Linear Relationships
In a non-linear function, the same input can lead to different outputs, indicating inconsistency in the relationship. For example, if you have an X input of 5 leading to multiple Y outputs, such as 7 and 8, this signifies a non-linear relationship. To identify non-linear behaviors, track outputs for consistent inputs and note any variations.
Step 3: Recognizing Patterns in Graphs
To determine if a function is linear, observe the changes in both X and Y values in relation to one another. A key indicator of a linear function is when increments in X result in predictable increments in Y. For instance, if increasing X by 1 leads to consistent, calculated changes in Y, this indicates a linear relationship. Ensure to analyze the graph for its structure and any noticeable straight lines to confirm linearity.