To figure out if a relationship is not a function, we first need to know what a function is in math.

A function is a special kind of relationship where every input (or starting value) has just one output (or ending value). In easier words, for every $x$ (input) we have, there should be one and only one $y$ (output) that goes with it.

How to Tell If Something Is Not a Function:

1. Vertical Line Test: This is a way to check a graph. If you draw a vertical line and it crosses the graph at more than one spot, then it is not a function. This rule works for every vertical line on the graph.

2. Unique Outputs: Check if one input has more than one output. For example, if $f(a) = b$ and $f(a) = c$ (where $b$ is not the same as $c$), then this relationship is not a function.

3. Examples of Non-Functions:

   • Circles: An equation like $x^2 + y^2 = r^2$ shows a circle. Here, one $x$ value can give you two different $y$ values (one above and one below the x-axis). So, circles don’t count as functions.
   • Piecewise Functions: Sometimes, certain piecewise definitions can also fail if they link the same input to different outputs.

4. Set Relationships: In a set of pairs that looks like $\{(1, 2), (1, 3)\}$, the first number '1' is connected to both 2 and 3. This means it’s not a function since '1' leads to two outputs.

Facts About Functions:

Studies show that many students find functions tricky. About 55% of 9th graders in the U.S. struggle with the vertical line test. Understanding what makes a function can really help students with their math skills and problem-solving.