Recognizing and Classifying Functions

What is a Function?

In mathematics, a function is a specific type of relationship between inputs and outputs. The golden rule of a function is that each input (usually $x$) must have exactly one output (usually $y$).

Let's look at a set of coordinate pairs: $\{(1,2), (2,3), (1,4)\}$. Is this a function?

• Look at the inputs (the first number in each pair): $1, 2, 1$.
• The input $1$ is paired with the output $2$, but it is also paired with the output $4$.
• Because the input $1$ gives two different outputs, this is not a function.

The Vertical Line Test

When you are looking at a graph, you can easily tell if it represents a function by using the vertical line test.

Imagine drawing straight, vertical lines down through the graph.

• If every vertical line touches the graph at exactly one point, the graph represents a function.
• If any vertical line touches the graph at more than one point, it is not a function. This happens because a single $x$-value (the vertical line) is giving multiple $y$-values (the intersection points).

Linear vs. Nonlinear Functions

Once you know a relationship is a function, you can classify it based on its rate of change.

Linear Functions

• Definition: A function with a constant rate of change.
• Graph: A straight line.
• Equation: Can be written in the form $y = mx + b$. (e.g., $y = 3x + 2$)

Nonlinear Functions

• Definition: A function where the rate of change is not constant.
• Graph: A curve or a line with bends.
• Equation: Contains exponents other than $1$, variables in the denominator, or other non-standard features.

Example: Is $y = x^2$ linear or nonlinear? Because the $x$ is squared, the rate of change will change as $x$ gets larger. The graph of $y = x^2$ is a curve (a parabola). Therefore, $y = x^2$ is a nonlinear function.