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If you drop a basketball off a 100-foot tower, its height after tt seconds is predictable. But writing y=100−16t2y = 100 - 16t^2 gets messy the moment you want to track heights at 1 second, 2 seconds, and 2.5 seconds in the same calculation.
In 1734, Swiss mathematician Leonhard Euler solved this clutter by inventing function notation, giving formulas a clean name and a designated input slot.

The Function Machine

A function is a mathematical rule that assigns every allowed input to exactly one output.
In the notation f(x)f(x), the letter ff names the function rule, while xx acts as the placeholder input — we read it aloud as "f of x."
What does it look like in practice when you swap that placeholder letter for an actual number?

Evaluating Step-by-Step

To evaluate a function, replace every occurrence of the input variable with empty parentheses, drop your chosen number inside, and simplify.
Let's find f(5)f(5) when f(x)=3x−8f(x) = 3x - 8.
First, write the shell with parentheses: f(5)=3(5)−8f(5) = 3(5) - 8. Multiplying gives 15−815 - 8, which simplifies to an output of 77.
Linear functions are simple enough — but what happens when powers and negative signs collide?