Question

Let $f : R \rightarrow R$ defined by $f ( x )=\cos ^{-1}(-\{- x \})$ where $\{x\}$ is fractional part function. Then which of the following is/are correct?

• A. f is many one but not even function.
• B. Range of f contains two prime numbers.
• C. f is aperiodic.
• D. Graph of f does not lie below x-axis

Answer 1

Answer: (A), (B), (D)

We have $f(x)=\cos ^{-1}(-\{-x\})$
$D _{ f }=$ all real
As $0 \leq\{- x \}<1 \forall x \in R$
$\Rightarrow -1<-\{- x \} \leq 0$
So $ R _{ f }=\left[\frac{\pi}{2}, \pi\right)$
Clearly, $f$ is neither even nor odd.
But $ f ( x +1)= f ( x ) \Rightarrow f$ is periodic with period 1.