To tell if an equation is a function, the key is to check whether each input (usually x) corresponds to exactly one output (usually y). Here are the main methods:

1. Algebraic Approach

• Solve the equation for y in terms of x.
• If for every x-value there is only one corresponding y-value, then the equation defines y as a function of x.
• If any x-value produces more than one y-value, it is not a function.
  For example, if solving yields y=±4−xy=\pm \sqrt{4-x}y=±4−x​, the plus-minus means two outputs for some x, so not a function

2. Vertical Line Test (Graphical)

• Graph the equation.
• Draw vertical lines through the graph.
• If any vertical line intersects the graph more than once, the relation is not a function.
• If every vertical line intersects at most once, it is a function

3. Conceptual Understanding

• A function is a relation where each input has at most one output.
• Think of it as a "machine" that takes an input and produces a single output.
• If you can list all (x, y) pairs and no x is repeated with different y's, it is a function

Summary Table

Method| What to Check| Outcome if True
---|---|---
Solve for y| One y for each x| Equation defines a function
Vertical Line Test| Vertical line intersects once max| Graph represents a function
Input-Output Pairing| Each input maps to one output| Relation is a function

In brief, an equation is a function if each input value corresponds to exactly one output value. The vertical line test on the graph or solving for y can confirm this