Let the functions $f: [-1,1] \rightarrow \mathbb{R}$ and $g: [-1,1] \rightarrow [-1,1]$ be defined by $f(x) = |2x-1| + |2x+1|$ and $g(x) = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Let $f \circ g: [-1,1] \rightarrow \mathbb{R}$ be the…

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Let the functions $f: [-1,1] \rightarrow \mathbb{R}$ and $g: [-1,1] \rightarrow [-1,1]$ be defined by $f(x) = |2x-1| + |2x+1|$ and $g(x) = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Let $f \circ g: [-1,1] \rightarrow \mathbb{R}$ be the composite function defined by $(f \circ g)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $[-1,1]$ at which $f \circ g$ is not continuous, and suppose $d$ is the number of points in the interval $[-1,1]$ at which $f \circ g$ is not differentiable. Then the value of $c+d$ is _____________

Aicharya · Accepted Answer

Graph of $g(x)$ is $f(x) = |2x - 1| + |2x + 1| = \begin{cases} -4x, & x \frac{1}{2} \end{cases}$ $f(g(x)) = \begin{cases} -4g(x), & g(x) \frac{1}{2} \end{cases}$ f g x = 2 , x ∈ - 1 , - 1 2 ∪ 0 , 1 2 4 x + 1 x ∈ - 1 2 , 0 4 x x ∈ 1 2 , 1 c = 1 at x = 0 d = 3 at x = - 1 2 , 0 , 1 2 , ∴ c + d = 4 . $g(x) = \{x\}$

Let the functions $f: [-1,1] \rightarrow \mathbb{R}$ and $g: [-1,1] \rightarrow [-1,1]$ be defined by $f(x)…

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