Question 70694
2022
The domain of the function f(x) = (cos−1x) / [x] is:
Answer: A. [−1, 0) ∪ {1} — ConceptA quotient is defined only where the numerator and the denominator are both defined and the denominator is not zero. So the domain of an expression of…
- A.
[−1, 0) ∪ {1}
- B.
[−1, 1]
- C.
[−1, 1)
- D.
None
Show answer & explanation
Correct answer: A
Concept
A quotient is defined only where the numerator and the denominator are both defined and the denominator is not zero. So the domain of an expression of the form p(x)/q(x) is the intersection of the domain of p and the domain of q, with every point where q equals zero removed.
Two standard facts supply those pieces here: cos−1t is defined only for −1 ≤ t ≤ 1, and the greatest-integer function [t], which returns the largest integer not exceeding t, equals 0 exactly when 0 ≤ t < 1.
Application
Numerator condition: cos−1x requires −1 ≤ x ≤ 1, so x must lie in [−1, 1].
Denominator condition: [x] must not be 0. Since [x] = 0 precisely for 0 ≤ x < 1, every x in [0, 1) has to be removed.
Intersection: removing [0, 1) from [−1, 1] leaves −1 ≤ x < 0 together with the single point x = 1, that is [−1, 0) ∪ {1}.
Cross-check
Testing sample values against both conditions:
x | cos−1x defined? | [x] | f(x) defined? |
|---|---|---|---|
−0.5 | Yes | −1 | Yes |
0 | Yes | 0 | No — division by zero |
0.9 | Yes | 0 | No — division by zero |
1 | Yes, cos−11 = 0 | 1 | Yes, f(1) = 0 |
1.5 | No | 1 | No |
Result
The expression is defined exactly on [−1, 0) ∪ {1}.