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Piecewise functions are a form of mathematical functions that have different expressions based on the interval of the input value, or 'pieces' of the domain. In simpler terms, think of piecewise functions as a chameleon changing colors to adapt to different surroundings; similarly, these functions change their rules depending on where in the number line the input value falls.

For the absolute value function given by the exercise, \( f(x) = |2x-5| \), this is represented by two linear functions: one where \( x \geq 2.5 \) and another for \( x < 2.5 \). By understanding where these changes occur, students can better visualize the graph and anticipate shifts in the function's behavior.

This concept is crucial for adequately graphing absolute value functions. Knowing this, students can decompose \( f(x) \) into its components and graph each piece accordingly to properly portray this piecewise nature on a coordinate plane.

Differentiability refers to a property of a function that determines whether a function has a derivative at a given point. Essentially, if you can draw a single, non-vertical tangent line at a point on the graph of the function, then it's differentiable at that point. Think about it as being able to smoothly slide along the curve without any jumps or sharp turns.

However, some functions have points where this isn't possible. For the absolute value function \( f(x) = |2x-5| \), this happens at \( x = 2.5 \), known as a 'cusp'. At a cusp, the graph changes direction so sharply that there's no single tangent line that touches the curve only at that point.

Appreciating where a function is not differentiable is key for understanding the underlying geometric structure and the behavior of the function at specific points. This becomes particularly important when dealing with optimization problems or when students need to compute derivatives as part of their calculus exercises.

Graphs of absolute value functions are distinctive due to their 'V' or inverted 'V' shapes. The absolute value function depicts distance from 0 on the number line, hence it is always non-negative. This results in graphs that reflect across the x-axis when the input value causes the inside of the absolute value to be negative.

For the function \( f(x) = |2x-5| \), the graph will have two arms—a positive slope arm for \( x \geq 2.5 \) and a negative slope arm for \( x < 2.5 \)—forming a sharp point at the vertex \( x = 2.5 \), the location of the graph's 'turnaround'.

By plotting these on a graph, students can see how the absolute value affects the linearity of the underlying expression, which is a cornerstone for understanding more complex absolute value graphs in future studies. The graph of an absolute value function is a fundamental visual aid that helps clarify the concept of piecewise-defined functions and their differentiability.