Question

Which of the following function(s) have no domain?

• A. $f(x)=\log _{x-1}\left(2-[x]-[x]^2\right)$ where $[x]$ denotes the greatest integer function.
• B. $g(x)=\cos ^{-1}(2-\{x\})$ where $\{x\}$ denotes the fractional part function.
• C. $h ( x )=\ln \ln (\cos x )$
• D. $f(x)=\frac{1}{\sec ^{-1}\left(\operatorname{sgn}\left(e^{-x}\right)\right)}$

Answer 1

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

(A) obviously $x>1$
for $1< x <2,[ x ]=1 \Rightarrow N =0$
for $x \geq 2, N <0 \Rightarrow$ no domain
(B) $0 \leq\{x\}<1$ hence $-1<-\{x\} \leq 0$
hence $1<2-\{x\} \leq 2$
$\Rightarrow \cos ^{-1}(2-\{ x \})$ does not exist. $]$