Sets, Relations & Functions
Functions
nta_abhyas_2025
Grade 11
Question:
For the function $f(z) = \sin(\lfloor z \rfloor) < \cos^{-1}(\lfloor z \rfloor)$, choose the correct option. (where $\lfloor \cdot \rfloor$ represents the greatest integer function)
Domain of f(z) ∈ [-1,1]
Range of f(z) contains exactly two elements
f(z) is an identity function
f(z) is a constant function
Step-by-Step Solution
Key Concept: A constant function outputs the same value for all inputs in its domain, so its range is a singleton set.
For any $x \in \mathbb{R}$, we know $\sin \pi x = 0$ only when $\pi x = n\pi$ for integer $n$. Thus $f(x) = 0$ for all $x \in [-1, 2)$ where $\sin \pi x = 0$. Since $f(x)$ takes the value $0$ for all valid $x$ in the domain, $f$ is a constant function with range $\{0\}$.
Correct Answer: $\{0\}$