Question

use the graph of the function \\(y = f(x)\\) to find all \\(x\\)-values where the function is differentiable. write your answer in interval notation. combine multiple intervals with the union symbol.

Explanation:

Identify points of non-differentiability from the graph

The graph of \(y = f(x)\) is a continuous, V-shaped curve with a sharp corner (cusp/vertex) at \(x = -4\).
A function is not differentiable at points where the graph has a sharp corner, cusp, vertical tangent, or discontinuity.
At \(x = -4\), the left-hand derivative is negative and the right-hand derivative is positive, meaning the derivative does not exist at \(x = -4\).

Determine the domain of differentiability

The function is defined and smooth everywhere else along its domain.
The arrows on the ends of the graph indicate that the function extends infinitely to the left and to the right.
Thus, the function is differentiable on all real numbers except at \(x = -4\).

Express the domain in interval notation

Excluding \(x = -4\) from the set of all real numbers \((-\infty, \infty)\) yields:

$$ (-\infty, -4) \cup (-4, \infty) $$

Answer:

\((-\infty, -4) \cup (-4, \infty)\)