Chapter exploration

What can go in, and what can come out?

Explore

Move the left boundary. Opening it excludes that endpoint. When the boundary passes the turning point, the minimum changes.

Move the boundary past the vertex and compare the input and output intervals.

Independent warm-up: f(x) = x² − 1, restricted to x ≥ k or x > k.

Left boundary k-1

Closed endpoint: the boundary is included unless the original domain excludes it.

Try an example

[-1, ∞)

[-1, ∞)

[-1, ∞)

(−∞, ∞)

These exclusions stay in force when a boundary changes. Arrows indicate continuation beyond the drawing window.

The allowed inputs contain 0. The minimum output −1 is reached, regardless of the left boundary being open or closed.

The graph shows a finite window; arrows and interval notation describe the full set. Decimal endpoints are rounded.