In this lesson
Inputs and outputs
Here's a trap nearly everyone falls into at least once: if it's drawn on a coordinate plane, it must be a function. A circle, a spiral, a scatter of points — they all graph beautifully, and most of them aren't functions at all. By the end of this lesson you'll have a ten-second test that separates the ones that are from the ones that aren't — and you'll know exactly what the test is checking for.
Start with the raw materials. When two quantities change together, one of them usually plays the role of input and the other the role of output. The input is the one you control or choose; the output is what changes in response.
Time is almost always an input. Distance, temperature, or count is almost always an output. The curriculum gives these roles formal names:
- The independent variable is the input. You choose its value first.
- The dependent variable is the output. Its value depends on the input you chose.
So "the time of day is the independent variable; the temperature is the dependent variable" reads as: choose a time, and the temperature follows from it. Switch the roles around and you get a different question, with temperature as input and time as output. That's sometimes meaningful and sometimes not.
What a relation is
A relation is the most general possible thing. It just says: some inputs are paired with some outputs. The pairing can be wild: multiple outputs for the same input, gaps, anything.
A relation can be written in four ways:
- A set of ordered pairs:
{(0, 1), (1, 3), (2, 5)}. - A table with input and output columns.
- A graph plotting each pair as a point or curve.
- A rule like
output = 2 × input + 1.
All four say the same thing. A relation is whatever set of pairings you write down.
Functions vs. relations
A function is a relation with one extra restriction:
Every function is a relation. Not every relation is a function. The extra restriction is the gate that separates the two.
With a function, knowing the input determines the output. There's no ambiguity. With a general relation, the same input can lead to different outputs and you have no rule for which to pick.
See it: scan the graph
The widget below lets you test relations visually — but it makes you call each one first. Pick a relation, commit to function or not a function, and only then does the scanner unlock. Drag it across the x-axis: if at every x it crosses the graph in exactly one place, the relation is a function. If at any x it crosses twice or more, it is not — and the scan will tell you whether your call survived.
Vertical line test
Drag the scanner along the x-axis. Count how many times the graph crosses the line at each x. If you ever count more than 1, the relation is NOT a function.
Before the scanner moves an inch — is this relation a function?
Hits at this x: 1 · Max hits seen: 0
This is the vertical line test. It's the visual proof of the function definition. Each vertical line corresponds to a single input x; the number of times it crosses the graph is the number of outputs the relation gives for that input. If any one of those counts is bigger than 1, the input has more than one output, and the relation is not a function.
Tables, graphs, ordered pairs
The vertical line test works on graphs. The "exactly one output per input" rule works on tables and ordered pairs too:
- Table: scan the input column. If any input value appears twice with different outputs, it's not a function.
- Ordered pairs: look at the first coordinates. If any first
coordinate appears twice with different second coordinates, it's
not a function.
{(1, 3), (2, 5), (1, 7)}is not a function: the input1has both3and7.
Same definition, three different representations. The vertical line test is what it looks like on a graph; the input-column check is what it looks like on a table.
Where it shows up in real life
A temperature-vs-time plot from an Alberta weather station is a function. Each time of day has exactly one recorded temperature. The air can't be two different temperatures at the same moment. The graph is the daily temperature curve, and it passes the vertical line test by construction.
If you tried the reverse, with temperature as input and time as output, you'd get a relation that's often not a function. The same temperature usually appears twice in a day: once as the thermometer rises, once as it falls. Knowing "it was 5 °C" doesn't tell you a single time; it tells you at least two.
A bus route schedule is a function: each scheduled stop has exactly one scheduled time. A circle on a map (showing all points 10 km from a starting point) is a relation but not a function: many of those points share an x-coordinate.
Functions are predictable: knowing the input determines the output. Real-world models that need predictability (forecast curves, scheduling, billing) are designed to be functions.
Worksheet
These aren't graded. Get them right, get them wrong. The goal is to apply the function definition cleanly to graphs, tables, and words.
Practice · Not graded
MA.7.FUN.1Practice the idea
01 / 08
Which of these statements correctly distinguishes a function from a relation?
Multiple choice: which statement correctly distinguishes a function from a relation?Show common mistakes
Student says
“Flags { (1, 2), (2, 4), (1, 2) } as 'not a function' because the input 1 appears twice.”
What it reveals
Treats any repetition of an input value as a function violation. The actual rule is more specific. A repeated input is only a problem when it's paired with DIFFERENT outputs.
Targeted response
Check the outputs too. (1, 2) and (1, 2) are the same pair; the input 1 still has one output, 2. The function definition forbids one input being paired with multiple DIFFERENT outputs, like (1, 2) and (1, 5). Listing the same pair twice is harmless.
Student says
“Treats every graph as a function. 'It's drawn on the coordinate plane, so it's a function.'”
What it reveals
Hasn't internalized the vertical line test as a check. The student is conflating 'a relation can be graphed' (true) with 'every graph is a function' (false).
Targeted response
Use the Vertical Line Test widget. Try the circle preset: at x = 0, the scanner crosses twice (once at y = 5, once at y = −5). One input, two outputs, so not a function. Some graphed relations pass the test; others don't. The graph is the relation; the test is what tells you whether it's also a function.
Student says
“Confuses independent and dependent variables. 'Distance is the independent variable because it controls how long it takes to drive.'”
What it reveals
Reversed the input/output direction. In the most natural reading, you choose how long to drive (input = time) and the distance follows from speed × time (output).
Targeted response
Ask: which one is chosen first, and which one responds? Time is chosen first (you decide how long to drive) and distance follows. So time is the independent variable; distance is the dependent variable. Some scenarios genuinely run the other way (e.g., 'how long does it take to cover 100 km?' makes distance the input), but read the problem carefully to decide.
Going further
The next lesson, Domain and Range, takes the function idea and asks two more questions about it: which inputs are allowed, and which outputs come out? Domain is the set of valid inputs; range is the set of resulting outputs. Both are attributes of the function itself.
In Grade 8 and beyond, you'll meet linear functions
(y = mx + b), then quadratic functions (y = ax² + bx + c), then
later still exponential and trigonometric. Every one of them is a
function (exactly one output per input) and every one of them has
its own domain and range. The vertical line test you used today
keeps working across all of them.