Question

Let $f(x) = \begin{cases} x-1; & x = \text{Even} \\ 2x; & x = \text{Odd} \end{cases}$, where $x \in N$. If for some $a \in N$, $f(f(f(a))) = 3$ and $m \& n$ respectively are the number of points in $(-a, a)$ at which $g(x) = |x^2 - 1|$ is not continuous and not differentiable, then $m + n$ is

• A. 0
• B. 2
• C. 4
• D. 6

Answer 1

Answer: (B)

2

Answer 2

The function $g(x) = |x^2 - 1|$ is continuous for all $x \in R$. It is not differentiable at points where $x^2 - 1 = 0$, which are $x = 1$ and $x = -1$. Therefore, $m=0$ (number of points of discontinuity) and $n=2$ (number of points of non-differentiability). For these points to be in the interval $(-a, a)$, we must have $a > 1$. Although the condition $f(f(f(a))) = 3$ does not yield a valid integer $a \in N$, if we assume the problem intends for $a > 1$, then $m+n = 0+2 = 2$.