Writing allowed inputs with intervals

What changes when an endpoint is included, and can infinity be included?

My prediction and reason

Predict whether x = 2 is allowed in 1/(x − 2), and whether x = 3 is allowed in √(x − 3). Explain why the two boundary decisions differ.

Evidence and observations

Explain or give a counterexample

Transfer

Write the domain of √(5 − x), then exclude x = 2 as an extra restriction. Show the included endpoint and excluded point correctly on a number line.

Exit task

Write the domain of 1/(x + 1), and say whether −1 is included. Write the domain of √(x + 1), and say whether −1 is included.

5 min predict boundary values; 12 min test and draw intervals; 8 min connect notation; 12 min paper practice; 3 min exit check.

What changes when an endpoint is included, and can infinity be included?

Predict whether x = 2 is allowed in 1/(x − 2), and whether x = 3 is allowed in √(x − 3). Explain why the two boundary decisions differ.

Evidence tasks

1. For 1/(x − 2), test x = 1, 2, 3. Mark 2 with an open point and shade both allowed sides.
2. For √(x − 3), test x = 2, 3, 4. Mark 3 with a closed point and shade the allowed side.
3. Write each domain in words, inequalities and interval notation. Check that all three forms make the same endpoint decision.

Reference reasoning

A real denominator cannot equal zero, so 1/(x − 2) allows all real x except 2. A real square root allows a zero radicand, so √(x − 3) allows x ≥ 3. Round brackets exclude a finite endpoint; square brackets include it. Infinity describes an unbounded direction, not an attainable endpoint, so it always takes a round bracket.

Transfer and exit task

Write the domain of √(5 − x), then exclude x = 2 as an extra restriction. Show the included endpoint and excluded point correctly on a number line.

Write the domain of 1/(x + 1), and say whether −1 is included. Write the domain of √(x + 1), and say whether −1 is included.

The reciprocal domain is (−∞, −1) ∪ (−1, ∞), excluding −1. The square-root domain is [−1, ∞), including −1.

• Uses ≠ 0 for a denominator and ≥ 0 for a square-root radicand.
• Matches interval brackets to the endpoint decisions and uses round brackets at infinity.