Analyzing Non-Differentiability Points for Composite Function

We are given the function $f(x) = |x - 2| + |x - 8|$ where $x$ belongs to the set of real numbers ($x \in R$). The task is to find the set of all values of $x$ for which the composite function $g(x) = f(f(x))$ is not differentiable.

Understanding the Function $f(x)$

The function $f(x)$ is defined using absolute values. A key property is that the absolute value function $|y|$ is not differentiable at $y=0$. This suggests that $f(x)$ might not be differentiable at points where the expressions inside the absolute values become zero, namely $x = 2$ and $x = 8$.

Let's break down $f(x)$ into different intervals:

• If $x < 2$: Both $(x-2)$ and $(x-8)$ are negative. So, $f(x) = -(x - 2) - (x - 8) = -x + 2 - x + 8 = -2x + 10$.
• If $2 \le x \le 8$: $(x-2)$ is non-negative, and $(x-8)$ is non-positive. So, $f(x) = (x - 2) - (x - 8) = x - 2 - x + 8 = 6$.
• If $x > 8$: Both $(x-2)$ and $(x-8)$ are positive. So, $f(x) = (x - 2) + (x - 8) = x - 2 + x - 8 = 2x - 10$.

We need to check the differentiability at the transition points $x=2$ and $x=8$:

• At $x=2$:
  • The derivative from the left (using $-2x + 10$) is $-2$.
  • The derivative from the right (using $6$) is $0$.
  • Since the left-hand derivative ($-2$) $\neq$ right-hand derivative ($0$), $f(x)$ is not differentiable at $x = 2$.
• At $x=8$:
  • The derivative from the left (using $6$) is $0$.
  • The derivative from the right (using $2x - 10$) is $2$.
  • Since the left-hand derivative ($0$) $\neq$ right-hand derivative ($2$), $f(x)$ is not differentiable at $x = 8$.

Thus, the function $f(x)$ is confirmed to be non-differentiable at $x = 2$ and $x = 8$. These points are potential candidates for where $g(x)$ is non-differentiable.

Determining the Range of $f(x)$

Understanding the output values (range) of $f(x)$ is crucial for analyzing the composite function $g(x) = f(f(x))$.

• For $x < 2$, $f(x) = -2x + 10$. As $x$ increases towards $2$, $f(x)$ decreases towards $6$. As $x$ decreases towards $-\infty$, $f(x)$ increases towards $\infty$. The range here is $(6, \infty)$.
• For $2 \le x \le 8$, $f(x) = 6$. The range here is just the value $6$.
• For $x > 8$, $f(x) = 2x - 10$. As $x$ increases from $8$, $f(x)$ increases from $6$ towards $\infty$. The range here is $(6, \infty)$.

Combining these, the complete range of $f(x)$ for $x \in R$ is $[6, \infty)$. This means $f(x)$ will always produce values greater than or equal to $6$.

Analyzing Non-Differentiability of $g(x) = f(f(x))$

For a composite function $g(x) = f(u)$, where $u = f(x)$, $g(x)$ fails to be differentiable at a point $x$ if:

1. The inner function $f(x)$ is not differentiable at $x$.
2. OR, the outer function $f(u)$ is not differentiable at the value $u = f(x)$.

Points from Condition 1: $f(x)$ is not differentiable

We have already determined that $f(x)$ is not differentiable at $x = 2$ and $x = 8$. Therefore, $g(x) = f(f(x))$ is also not differentiable at these two points.

Points from Condition 2: $f(u)$ is not differentiable at $u=f(x)$

The outer function $f(u)$ is not differentiable when $u=2$ or $u=8$. We need to find the values of $x$ such that $f(x)$ equals $2$ or $8$.

• Check $f(x) = 2$: From the range analysis, we know that $f(x)$ is always greater than or equal to $6$ ($f(x) \in [6, \infty)$). Therefore, $f(x)$ can never be equal to $2$. There are no $x$ values satisfying $f(x)=2$.
• Check $f(x) = 8$: We look at our piecewise definition of $f(x)$:
  • If $x < 2$, $f(x) = -2x + 10$. We set this to $8$: $ -2x + 10 = 8 $ $ -2x = -2 $ $ x = 1 $ This value $x=1$ is indeed less than $2$, so it's a valid solution. At $x=1$, $f(1)=8$, and $f(u)$ is not differentiable at $u=8$. Thus, $g(x)$ is not differentiable at $x=1$.
  • If $2 \le x \le 8$, $f(x) = 6$. Since $6 \neq 8$, there are no solutions in this interval.
  • If $x > 8$, $f(x) = 2x - 10$. We set this to $8$: $ 2x - 10 = 8 $ $ 2x = 18 $ $ x = 9 $ This value $x=9$ is indeed greater than $8$, so it's a valid solution. At $x=9$, $f(9)=8$, and $f(u)$ is not differentiable at $u=8$. Thus, $g(x)$ is not differentiable at $x=9$.
  So, the values of $x$ for which $f(x)=8$ are $x=1$ and $x=9$.

Combining All Points of Non-Differentiability

To find all points where $g(x)$ is not differentiable, we combine the results from both conditions:

• Points where $f(x)$ itself is not differentiable: $\{2, 8\}$.
• Points $x$ where $f(x)$ equals a non-differentiable point of $f$ (i.e., $f(x)=2$ or $f(x)=8$): $\{1, 9\}$.

The set of all values of $x$ where $g(x) = f(f(x))$ is not differentiable is the union of these sets: $\{1, 2, 8, 9\}$.

Final Conclusion

The set containing all values of $x$ at which the function $g(x) = f(f(x))$ is not differentiable is $\{1, 2, 8, 9\}$.