Find the domain of f(x) = sin⁻¹(2x − 3) + cos⁻¹(x − 1).
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: f(x) = sin⁻¹(2x − 3) + cos⁻¹(x − 1). To find: all x for which f(x) is defined.
2 · Write the rule. sin⁻¹ u and cos⁻¹ u are defined only for −1 ≤ u ≤ 1. For a sum, x must satisfy both conditions.
3 · Domain of the first part.
−1 ≤ 2x − 3 ≤ 1
Add 3: 2 ≤ 2x ≤ 4
Divide by 2: 1 ≤ x ≤ 2
4 · Domain of the second part.
−1 ≤ x − 1 ≤ 1
Add 1: 0 ≤ x ≤ 2
5 · Take the common values. [1, 2] ∩ [0, 2] = [1, 2].
6 · Check and box the answer. At x = 1: sin⁻¹(−1) + cos⁻¹(0) = −π/2 + π/2 = 0. At x = 2: sin⁻¹(1) + cos⁻¹(1) = π/2 + 0 = π/2. Both are defined. Domain = [1, 2].
Scoring tip: Show each inequality line by line. Examiners give marks for the working, not just the final interval.
