Domain and Range as Valid Inputs and Outputs (F 506)

A function is a rule, but not every number is allowed to go into that rule. Domain is the set of inputs the function accepts without breaking math — no dividing by zero, no square-rooting a negative. Range is just the leftover pile of outputs those inputs produce. Every function comes with both because a rule without restrictions would try to process impossible numbers, like √(-4) or 1/0, and the whole thing falls apart.

On a graph, domain is the horizontal shadow the curve casts onto the x-axis: scan left to right and note where the graph starts and stops. Range is the vertical shadow onto the y-axis: scan bottom to top. Endpoints matter. A closed, filled-in dot means that exact point belongs to the graph, so its coordinate counts in the domain or range. An open, hollow dot means the graph approaches that point but never actually lands there, so that coordinate gets excluded — you'd write < or > instead of ≤ or ≥.

This connects directly to vertex form, a(x - h)² + k, which hands you the range immediately. The vertex (h, k) is the highest or lowest point of the parabola. If a is positive, the parabola opens upward and the vertex is a minimum, so range is y ≥ k. If a is negative, it opens downward and the vertex is a maximum, so range is y ≤ k. Domain for any parabola is always all real numbers, since x can be anything — only the range depends on the shape.