Functions
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What is a Function?

Discover functions through a real-world problem — from dots to lines to solutions

Functions are everywhere — your GPS turns location into distance, a vending machine turns a button press into a snack, a thermometer turns temperature into a number. What if you could predict the future? If Alex walks at 4 miles per hour, you know he'll be at 8 miles after 2 hours, 12 miles after 3 hours. That predictable pattern — where every input (time) gives exactly one output (distance) — is called a function.

In this lesson, we'll start with a simple walking problem and build up to the mathematical definition of a function. You'll see how data points form a line, how that line predicts values you never measured, and how drawing two lines can solve a problem without any algebra.

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FAQ

What is a function in math?
A function is a rule that assigns exactly one output to each input. For example, f(x) = 4x takes any number x and gives back 4 times x. The key requirement: each input produces exactly one output — never two, never zero.
What is the vertical line test?
The vertical line test checks whether a graph represents a function. Draw a vertical line anywhere on the graph. If it hits the curve at exactly one point, it's a function. If it hits two or more points (like on a circle), it's not a function — because one input would give multiple outputs.
What are domain and range?
The domain is the set of all allowed inputs (x-values). The range is the set of all possible outputs (y-values). For example, if f(x) = 4x models walking (time → distance), the domain is x ≥ 0 (you can't walk negative hours) and the range is y ≥ 0 (distance is never negative).
How do you solve a problem graphically?
Graph the function that models the behavior, then graph the goal as a horizontal line. Where the two lines intersect is your answer. For example, to find when someone walking at 4 mph covers 20 miles: graph y = 4x and y = 20, and read the x-value where they cross (x = 5 hours).

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