# Please let me know the answers for 11, 12 and 13 questions in passage 1

Abhishek Singh
105 Points
one year ago
Clearly
\begin{align*} f(x) &= \sqrt{\frac{1+Cos2x}{2}}\\ &=\sqrt{Cos^2x} \\ &= |Cos(x)| \end{align*}
Now Draw the graph of |Cos(x)| and check the injectivity by Horizontal line test and see it is injective for option (C)

12) f(x) = |x|, Clearly it becomes onto for optiond (D) by graph
13) For explicitly writing the function, check that
if x is integer f(x) = [x] + [-x] = x+ (-x) = 0
if x is non integer f(x) = [x] + [-x] = [x] + (-[x] -1) = -1

$f(x) \begin{cases} 0& \text{ if } x= integer\\ -1&\text{ if } x= non-integer \end{cases}$
Its range is {0,-1} which is not in options