Question

Let $f\left(x\right)=\frac{\left[x\right]+1}{\left\{x\right\}+1},forf:[0,\frac{5}{2})\to (\frac{1}{2},3]$ where [x] denotes greatest integer function and {x} denotes fractional part function, then which of the following is/are true?

• A. f(x) is one - one function
• B. f(x) is onto but not one - one function
• C. f(x) is a bijective function
• D. f(x) is symmetric about $x=\frac{5}{4}$

Answer 1

Answer: (A), (C)

f(x) is a bijective function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{x+1}& 0\le x<1\\ \frac{2}{x}& 1\le x<2\\ \frac{3}{x-1}& 2\le x<\frac{5}{2}\end{array}\right\}$

Clearly f(x) is one - one, onto function and not symmetric about $x=\frac{5}{4}.$