An equation represents a function if every input xxx gives exactly one output yyy.

Core idea in plain language

  • Think of a function as a machine: you put in one xxx, you get back only one yyy.
  • If some xxx could produce two different yyy-values, then the equation is not a function.

Example:

  • y=x+3y=x+3y=x+3 is a function (each xxx gives one yyy).
  • x2+y2=1x^2+y^2=1x2+y2=1 is not a function of xxx because many xxx-values give two possible yyy-values (top and bottom of the circle).

Quick test 1: Solve for yyy

When you can, solve the equation for yyy.

  1. Isolate yyy in terms of xxx.
  2. Check how many yyy-values you get for a single xxx.
  • If you get one expression for yyy, it’s a function.
    • Example: y=2x−5y=2x-5y=2x−5 → one yyy for each xxx.
  • If you get ± \pm ± or multiple branches, usually not a function.
    • Example: y2=xy^2=xy2=x → y=±xy=\pm\sqrt{x}y=±x​ (two yyy-values for each x>0x>0x>0) → not a function.
* Example: x2+y2=1x^2+y^2=1x2+y2=1 → y=±1−x2y=\pm\sqrt{1-x^2}y=±1−x2​ → not a function.

Rule of thumb: if solving gives only one yyy for each xxx, the equation defines yyy as a function of xxx.

Quick test 2: Vertical Line Test (graph idea)

If you imagine or actually graph the equation:

  • Draw (or imagine) vertical lines x=constantx=\text{constant}x=constant.
  • If every vertical line hits the graph at at most one point , it is a function of xxx.
  • If any vertical line hits the graph twice or more , it is not a function of xxx.

Examples:

  • Lines like y=3x+1y=3x+1y=3x+1 and parabolas like y=x2y=x^2y=x2 pass the vertical line test → functions.
  • Circles like x2+y2=4x^2+y^2=4x2+y2=4 fail it (a vertical line cuts top and bottom of the circle) → not functions.
  • Vertical lines like x=2x=2x=2 are not functions of xxx (infinite yyy-values for one xxx).

Quick test 3: Think in ordered pairs

Imagine all the solutions as pairs (x,y)(x,y)(x,y).

  • If no xxx-value repeats with different yyy-values, it’s a function.
  • If some xxx appears with two different yyy’s, it is not a function.

Example:

  • (1,2),(2,3),(3,4)(1,2),(2,3),(3,4)(1,2),(2,3),(3,4) → function (each input has one output).
  • (1,2),(1,−2),(2,3)(1,2),(1,-2),(2,3)(1,2),(1,−2),(2,3) → not a function (input 1 has two outputs).

This is just the vertical line test written in list form.

Typical patterns to recognize

  • Always functions of xxx:
    • Linear: y=mx+by=mx+by=mx+b
    • Polynomials: y=x2−1y=x^2-1y=x2−1, y=x3+2xy=x^3+2xy=x3+2x
    • Exponential: y=2xy=2^xy=2x
    • Absolute value: y=∣x∣y=|x|y=∣x∣
      Each xxx gives one yyy.
  • Usually not functions of xxx:
    • Equations with y2y^2y2, ∣y∣|y|∣y∣, or ±\pm ± when solved: e.g. x2+y2=1x^2+y^2=1x2+y2=1, ∣y∣=4−x|y|=4-x∣y∣=4−x.

These tend to give two yyy-values for some xxx.

Mini story to lock it in

Imagine you’re running a help desk where each student ID (your input) must map to exactly one locker number (your output).

  • If some student ID is given two lockers, your system is broken → that’s like an equation that is not a function.
  • If every student ID has one and only one locker, your system is working → that’s a function.

Fast checklist: “Is this equation a function?”

Ask yourself:

  1. Can I solve for yyy as a single expression in xxx?
  2. If I plug in a specific xxx, do I ever get more than one possible yyy?
  3. If I picture the graph, would any vertical line hit it more than once?
  • If your answer is no to extra yyy-values or double hits, it’s a function.
  • If yes , it is not a function.

TL;DR:
An equation is a function (of xxx) when each xxx leads to exactly one yyy, which you can check by solving for yyy, using the vertical line test, or checking that no xxx-value is paired with two different yyy-values.

Information gathered from public forums or data available on the internet and portrayed here.