Problems Based on Piecewise Defined Functions

Piecewise defined functions are functions that have different expressions for different intervals of their domain. These functions can pose unique challenges when it comes to analyzing their properties, such as continuity, differentiability, and integrability. Understanding how to work with piecewise functions is crucial for solving problems in calculus and other areas of advanced mathematics.

Understanding Piecewise Functions

A piecewise function is defined by multiple sub-functions, each of which applies to a certain interval of the domain. The general form of a piecewise function can be written as:

$$ f(x) = \begin{cases} f_1(x) & \text{if } a \leq x < b \ f_2(x) & \text{if } b \leq x < c \ \vdots \ f_n(x) & \text{if } z \leq x \leq w \end{cases} $$

where $f_1(x), f_2(x), \ldots, f_n(x)$ are the different sub-functions that define $f(x)$ over specific intervals.

Important Points to Consider

When dealing with piecewise functions, there are several key points to consider:

• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values the function can produce.
• Continuity: A function is continuous at a point if the limit as it approaches the point from both sides equals the function's value at that point.
• Differentiability: A function is differentiable at a point if it has a defined derivative at that point, which usually requires the function to be continuous at that point.

Table of Differences and Important Points

Property	Description	Piecewise Function Consideration
Domain	Set of all input values	Each sub-function has its own domain interval.
Range	Set of all output values	Determined by the outputs of all sub-functions.
Continuity	Function has no breaks, jumps, or holes	Check continuity at the endpoints of each interval.
Differentiability	Function has a defined derivative	Check differentiability at the endpoints of each interval.

Examples of Piecewise Functions

Let's look at some examples to illustrate how to analyze piecewise functions.

Example 1: Continuity

Consider the piecewise function:

$$ f(x) = \begin{cases} x^2 & \text{if } x < 1 \ 2x - 1 & \text{if } x \geq 1 \end{cases} $$

To determine if $f(x)$ is continuous at $x = 1$, we need to check the following:

1. $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1$
2. $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x - 1) = 1$
3. $f(1) = 2(1) - 1 = 1$

Since all three values are equal, $f(x)$ is continuous at $x = 1$.

Example 2: Differentiability

Consider the same function as above:

$$ f(x) = \begin{cases} x^2 & \text{if } x < 1 \ 2x - 1 & \text{if } x \geq 1 \end{cases} $$

To determine if $f(x)$ is differentiable at $x = 1$, we need to check the following:

1. The left-hand derivative: $\lim_{h \to 0^-} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^-} \frac{(1 + h)^2 - 1}{h} = 2$
2. The right-hand derivative: $\lim_{h \to 0^+} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^+} \frac{2(1 + h) - 1 - 1}{h} = 2$

Since both the left-hand and right-hand derivatives are equal, $f(x)$ is differentiable at $x = 1$.

Example 3: Integrability

Consider the function:

$$ f(x) = \begin{cases} 3 & \text{if } 0 \leq x < 2 \ x + 1 & \text{if } 2 \leq x \leq 4 \end{cases} $$

To find the integral of $f(x)$ from 0 to 4, we split the integral into two parts:

$$ \int_0^4 f(x) \, dx = \int_0^2 3 \, dx + \int_2^4 (x + 1) \, dx $$

Evaluating the integrals, we get:

$$ \int_0^2 3 \, dx = [3x]_0^2 = 6 $$

$$ \int_2^4 (x + 1) \, dx = \left[\frac{1}{2}x^2 + x\right]_2^4 = (8 + 4) - (2 + 2) = 8 $$

Therefore, the integral of $f(x)$ from 0 to 4 is $6 + 8 = 14$.

In conclusion, working with piecewise functions requires careful consideration of each sub-function's behavior at the boundaries of their respective intervals. Continuity and differentiability must be checked at these points, and integrals may need to be broken into separate parts corresponding to each interval. Understanding these concepts is essential for solving problems involving piecewise defined functions.