Functions & Notation: f(x), Domain, Range, and the Vertical Line Test
A function is a rule that takes one input and gives one output. Master f(x) notation, domain/range, and the vertical line test — the foundation for every equation and graph in Algebra 1.
A function is a rule, not a number
If “solving for x” gives you one number, a function gives you a rule — one that turns any input into a single output. Once you understand f(x) notation, function questions become a one-step plug-in.
A function is a machine. You drop in x, the rule processes it, and out pops f(x). Same x in → same f(x) out, every time.
Evaluating f(x) at a value
To evaluate a function at a number, simply substitute that number for x.
Reverse direction: solve for the input
Sometimes you get the output and have to find the input.
The vertical line test
How do you tell if a graph is a function? Drop a vertical line anywhere on it. If your vertical line ever crosses the graph at more than one point, it's not a function.
Domain & range
- Domain
- All x-values the function accepts (inputs). Read the graph horizontally.
- Range
- All y-values the function produces (outputs). Read the graph vertically.
- You cannot divide by zero → exclude any x that makes a denominator zero.
- You cannot take an even root of a negative → expressions under √ must be ≥ 0.
3-second recap
- f(x) means “the rule, evaluated at x” — substitute and simplify.
- Vertical line test: function = at most one y per x.
- Domain = x-inputs allowed; range = y-outputs produced.
- Watch out for division by zero and even roots of negatives when finding domain.