Piecewise Functions

Piecewise Functions

A piecewise function is a function that is defined by different expressions or formulas for different intervals of its domain. Each "piece" of the function applies to a specific part of the domain, and the function is evaluated differently depending on which part of the domain $x$ belongs to.

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1. Definition of Piecewise Functions

A piecewise function is a function that has different rules (or formulas) for different intervals of its domain. The function is defined by multiple expressions, each of which applies to a specific range of input values.

A piecewise function can be written in the following general form:

$$f(x) = \begin{cases} f_1(x) & \text{if } x \in I_1 \\ f_2(x) & \text{if } x \in I_2 \\ \vdots \\ f_n(x) & \text{if } x \in I_n \end{cases}$$

Where:

• $f_1(x), f_2(x), \ldots, f_n(x)$ are different expressions or formulas for the function, each applying to a specific interval $I_1, I_2, \ldots, I_n$ of the domain.

Each expression applies to a specific interval of the input $x$, and the domain is divided into pieces, hence the name "piecewise function."

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2. How to Read Piecewise Functions

To evaluate a piecewise function, you must:

1. Identify the interval where the input $x$ lies.
2. Use the corresponding formula for that interval to calculate the value of the function.

For example, if the piecewise function is defined as:

$$f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x + 1 & \text{if } x > 1 \end{cases}$$

To evaluate $f(x)$ for a particular $x$:

• If $x \leq 1$, use the formula $f(x) = x^2$.
• If $x > 1$, use the formula $f(x) = 2x + 1$.

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3. Example of a Piecewise Function

Consider the piecewise function:

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

• For $x < -1$, use the formula $f(x) = 3x + 2$.
• For $-1 \leq x < 2$, use the formula $f(x) = -x + 1$.
• For $x \geq 2$, use the formula $f(x) = x^2$.

Example Evaluations:

• If $x = -2$, since $-2 < -1$, we use $f(x) = 3x + 2$:

  $$f(-2) = 3(-2) + 2 = -6 + 2 = -4$$

• If $x = 0$, since $-1 \leq 0 < 2$, we use $f(x) = -x + 1$:

  $$f(0) = -(0) + 1 = 1$$

• If $x = 3$, since $x \geq 2$, we use $f(x) = x^2$:

  $$f(3) = (3)^2 = 9$$

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4. Graphing Piecewise Functions

To graph a piecewise function, follow these steps:

1. Identify the intervals where each piece of the function is defined.
2. Plot each piece of the function on its respective interval.
3. Check for continuity: Ensure that there are no gaps or jumps between the pieces at the boundary points.

• Closed endpoints (e.g., $x = -1$ in $-1 \leq x < 2$) mean the point is included in the interval.
• Open endpoints (e.g., $x = 2$ in $x \geq 2$) mean the point is not included in the interval.
• If the function transitions from one formula to another at the boundary, check if the limit from both sides matches to determine whether there is a jump discontinuity or if the function is continuous.

Example:

For the piecewise function:

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

• For $x < 0$, the function is a straight line with slope 2, intersecting the y-axis at $(0, 1)$.
• For $x \geq 0$, the function is a parabola opening upwards, starting at the origin.

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5. Special Types of Piecewise Functions

1. Step Functions: These are piecewise functions where each piece is a constant. For example:

   $$f(x) = \begin{cases} 1 & \text{if } x < 0 \\ 2 & \text{if } x \geq 0 \end{cases}$$

   This function "jumps" from 1 to 2 at $x = 0$.

2. Absolute Value Functions: These are often expressed as piecewise functions, as the absolute value function has two different expressions for positive and negative inputs. For example:

   $$f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

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6. Continuity and Discontinuity in Piecewise Functions

• A piecewise function can be continuous or discontinuous at the boundaries between pieces.
• Continuity: The function is continuous at a boundary if the two one-sided limits match at that point.
• Discontinuity: A function has a discontinuity (often a "jump") at a boundary if the one-sided limits do not match.

Example of a Continuous Piecewise Function:

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

In this case, the function is continuous because both pieces are the same, so there's no jump or break at $x = 2$.

Example of a Discontinuous Piecewise Function:

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

In this case, there's a jump discontinuity at $x = 2$, as $\lim_{x \to 2^-} f(x) = 3$ and $\lim_{x \to 2^+} f(x) = -1$.

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7. Summary of Key Concepts

Concept	Description
Piecewise Function	A function defined by different expressions on different intervals.
How to Evaluate	Identify the correct interval for $x$, then apply the corresponding expression.
Graphing Piecewise Functions	Plot each piece of the function over its interval and check for continuity.
Continuity	A piecewise function is continuous at a boundary if the one-sided limits match.
Discontinuity	A piecewise function has a jump discontinuity if the one-sided limits do not match.

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Conclusion

Piecewise functions are used to describe situations where a rule or formula changes based on the input value. They are essential for modeling real-world phenomena that behave differently in different conditions, such as tax brackets, shipping rates, and certain physical processes. Understanding how to define, evaluate, and graph piecewise functions is key to solving problems involving such situations.