ACT Preparation / Unit 10 ACT 0909 worked example, 60 to 90 seconds
Finding a Domain
The idea
To find a domain from a formula, work in three steps. First, start from all real numbers. Second, look for each restriction: a denominator that could be zero, an even root that could hold a negative, and any limit the situation imposes. Third, write each restriction as an inequality or an excluded value, solve it, and combine them all.
In practice
Worked example: h(x) = √(6 − 2x)/(x + 1). The root needs 6 − 2x ≥ 0, so x ≤ 3. The denominator needs x + 1 ≠ 0, so x ≠ −1. Combined: x ≤ 3 and x ≠ −1, or in interval notation, (−∞, −1) ∪ (−1, 3].
Where people slip
Solve the root's inequality carefully: 6 − 2x ≥ 0 gives −2x ≥ −6, and dividing by −2 reverses the sign, so x ≤ 3, not x ≥ 3.
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