piecewise function examples with answers pdf

2 min read 31-01-2025

piecewise function examples with answers pdf

Piecewise functions, those mathematical chameleons that change their behavior depending on the input, can seem daunting at first. But with a little practice and the right examples, they become much clearer. This guide provides a range of piecewise function examples with detailed solutions, helping you master this important concept. We'll cover various scenarios, from simple to more complex, ensuring you develop a strong understanding.

What is a Piecewise Function?

A piecewise function is defined by multiple sub-functions, each applicable over a specific interval of the domain. Think of it as a function with different "pieces," each piece taking control for a particular section of the x-axis. These pieces are defined using inequalities or specific values of x. The key is understanding which "piece" of the function to use based on the input value.

The general form looks like this:

f(x) = {
  g(x), if a ≤ x < b
  h(x), if b ≤ x < c
  i(x), if x ≥ c
}

Here, g(x), h(x), and i(x) are different functions, each active within its specified interval.

Piecewise Function Examples with Solutions

Let's dive into some examples, progressing from basic to more advanced scenarios.

Example 1: A Simple Piecewise Function

Let's define a function:

f(x) = {
  x + 1, if x < 2
  x - 1, if x ≥ 2
}

Find f(1) and f(3).

f(1): Since 1 < 2, we use the first piece: f(1) = 1 + 1 = 2
f(3): Since 3 ≥ 2, we use the second piece: f(3) = 3 - 1 = 2

Example 2: A Piecewise Function with an Absolute Value

Absolute value functions are often incorporated into piecewise functions. Consider this example:

f(x) = |x|

This can be rewritten as a piecewise function:

f(x) = {
  -x, if x < 0
  x, if x ≥ 0
}

Find f(-2) and f(5).

f(-2): Since -2 < 0, we use the first piece: f(-2) = -(-2) = 2
f(5): Since 5 ≥ 0, we use the second piece: f(5) = 5

Example 3: A More Complex Piecewise Function

This example involves a function with three pieces:

f(x) = {
  x², if x ≤ -1
  2x, if -1 < x < 1
  x + 1, if x ≥ 1
}

Find f(-2), f(0), and f(2).

f(-2): Since -2 ≤ -1, we use the first piece: f(-2) = (-2)² = 4
f(0): Since -1 < 0 < 1, we use the second piece: f(0) = 2(0) = 0
f(2): Since 2 ≥ 1, we use the third piece: f(2) = 2 + 1 = 3

Example 4: Evaluating a Piecewise Function Graphically

Sometimes you'll encounter piecewise functions represented graphically. The graph will clearly show the different parts of the function and their corresponding intervals. Evaluating the function at a given point simply involves identifying the correct section of the graph based on the x-value.

Tips for Working with Piecewise Functions

Carefully examine the intervals: Pay close attention to the inequalities defining each piece of the function. The correct piece must be selected based on the input value.
Substitute correctly: Once the correct piece is identified, substitute the input value into that specific function.
Check your work: It's easy to make a mistake choosing the wrong piece. Double-check your work to ensure accuracy.

This guide provides a solid foundation for understanding and working with piecewise functions. By practicing with these examples and exploring additional problems, you will develop the skills necessary to confidently tackle more complex piecewise function scenarios. Remember, practice is key! The more you work with these functions, the more intuitive they become.