91Ó°ÊÓ

The absolute value function, denoted as \(|x|\), is a mathematical function that gives the non-negative value of \(x\). It is defined as \(|x| = x\) when \(x \geq 0\) and \(|x| = -x\) when \(x < 0\). This function is crucial because it changes the behavior of the numerator or denominator in expressions, affecting differentiability.
For example, in the function \((f(x) = \frac{x}{1+|x|})\), the absolute value \(|x|\) alters the expression based on whether \(x\) is positive or negative:

• When \(x < 0\), \(|x| = -x\), translating the function to \(f(x) = \frac{x}{1-x}\).
• When \(x \geq 0\), \(|x| = x\), making \(f(x) = \frac{x}{1+x}\).

This change must be considered when evaluating where a function is continuous or differentiable, especially at the points where \(|x|\) switches values.

Piecewise functions are those defined by different expressions over different intervals of the domain. In the exercise, the function \((f(x) = \frac{x}{1+|x|})\) operates differently depending on whether \(x\) is positive or negative. Thus, it's evaluated as a piecewise function:

• For \(x < 0,\) the function is \(f(x) = \frac{x}{1-x}.\)
• For \(x \geq 0,\) it changes to \(f(x) = \frac{x}{1+x}.\)

Piecewise functions are instrumental in mathematics because they can model real-world scenarios where behavior changes across different conditions. The differentiability of a piecewise function can vary from one piece to another; hence, each piece must be assessed separately.
In this example, evaluating each part separately allowed us to determine that the function is not differentiable at \(x=0\), where these pieces converge but differ in slope. This illustrates a key challenge with piecewise functions: ensuring transitions between pieces are smooth to achieve differentiability.

Left and right derivatives help assess whether a function is differentiable at a certain point. A function is said to be differentiable at a point if both the left and right derivatives at that point exist and are equal.
When dealing with the function \((f(x) = \frac{x}{1+|x|})\), it's important to evaluate it from both sides:

• Left derivative: For \(h \to 0^-\), the approach is from the left side. Calculate the limit as \((\frac{f(h) - f(0)}{h})\) approaches there.
• Right derivative: For \(h \to 0^+\), the approach is from the right side. Similarly, compute the limit of \((\frac{f(h) - f(0)}{h})\) as \(h\) goes to zero from the right.

In our exercise, these derivatives yielded different values, meaning the function is not differentiable at \(x=0\). This discrepancy is a classic signal in calculus that a function lacks smoothness or continuity at a point, highlighting the importance of checking both left and right limits when verifying differentiability.