5-2 Additional Practice Piecewise Defined Functions

5+2 Additional Practice Piecewise Defined Functions: Mastering the Challenges

Piecewise defined functions, those mathematical chameleons that shift their behavior depending on the input value, can initially seem daunting. However, with practice and a structured approach, understanding and working with them becomes significantly easier. This article delves into five core examples of piecewise functions, followed by two additional, more challenging problems designed to solidify your understanding and build your problem-solving skills. We'll explore the intricacies of each function, focusing on graphing techniques, domain and range determination, and evaluating function values at specific points.

Understanding Piecewise Functions

Before we dive into the examples, let's briefly recap the definition. A piecewise defined function is a function defined by multiple sub-functions, each applicable to a specific interval of the domain. These intervals are typically defined using inequalities or specific values. The key is understanding which sub-function to use based on the input value (x).

The general structure of a piecewise function looks like this:

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

Where g(x), h(x), and i(x) are different functions, and a, b, and c define the intervals for each sub-function.

Example 1: The Absolute Value Function

The absolute value function, |x|, is a classic example of a piecewise function. It can be defined as:

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

Graphing: The graph forms a "V" shape, with the vertex at (0,0). For x < 0, the graph is a line with a slope of -1, and for x ≥ 0, it's a line with a slope of 1.

Domain and Range: The domain is all real numbers (-∞, ∞), and the range is all non-negative real numbers [0, ∞).

Example 2: A Step Function

Step functions represent a discontinuous jump in the function's value at specific points. Consider the following:

f(x) = {
  1, if x < 2
  3, if 2 ≤ x < 5
  5, if x ≥ 5
}

Graphing: This will produce a step-like graph. The function value remains constant within each interval and jumps at x = 2 and x = 5.

Domain and Range: The domain is all real numbers (-∞, ∞). The range is a discrete set {1, 3, 5}.

Example 3: A Combination of Linear and Quadratic Functions

This example shows how different types of functions can be combined within a piecewise definition.

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

Graphing: This will have a parabolic curve (x² - 1) for x < 1 and a straight line (2x + 1) for x ≥ 1. Notice the point (1,3) where the two parts connect.

Domain and Range: The domain is all real numbers (-∞, ∞). The range requires a more careful consideration; analyzing the individual functions shows that the range is also all real numbers (-∞, ∞).

Example 4: A Piecewise Function with a Hole

This example introduces the concept of a "hole" or removable discontinuity.

f(x) = {
  (x² - 4) / (x - 2), if x ≠ 2
  5, if x = 2
}

Graphing: Notice that (x² - 4) / (x - 2) simplifies to (x + 2) when x ≠ 2. Thus, the graph is a straight line y = x + 2, except at x = 2 where there is a hole. The function is defined to be 5 at x = 2, filling the hole with a single point.

Domain and Range: The domain is all real numbers (-∞, ∞). The range is all real numbers except for 4 (because the simplified line y = x + 2 would have a value of 4 at x = 2, but the piecewise definition overrides this with a value of 5).

Example 5: A More Complex Piecewise Function

This example showcases a piecewise function incorporating multiple sub-functions and intervals.

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

Graphing: This combines a line segment (-x + 2), a parabolic segment (x²), and a constant segment (4). Careful attention needs to be paid to the connecting points at x = -1 and x = 2.

Domain and Range: The domain is all real numbers (-∞, ∞). The range, after analyzing each interval, will be (-∞, 4].

Additional Challenge Problem 1: Evaluating a Piecewise Function

Let's consider this piecewise defined function:

f(x) = {
  √(x + 5), if -5 ≤ x < 2
  x - 1, if 2 ≤ x < 5
  x² - 8, if x ≥ 5
}

Evaluate f(-5), f(0), f(2), f(4), and f(7).

Solution:

• f(-5) = √(-5 + 5) = √0 = 0
• f(0) = √(0 + 5) = √5
• f(2) = 2 - 1 = 1
• f(4) = 4 - 1 = 3
• f(7) = 7² - 8 = 41

Additional Challenge Problem 2: Finding the Domain and Range of a Complex Piecewise Function

Consider the piecewise function:

f(x) = {
  1/(x+3), if x < -3
  x² - 9, if -3 ≤ x < 3
  √(x-3), if x ≥ 3
}

Determine the domain and range of f(x).

Solution:

Domain:

• The first piece, 1/(x+3), is undefined at x = -3.
• The second piece, x² - 9, is defined for -3 ≤ x < 3.
• The third piece, √(x-3), is defined for x ≥ 3.
• Therefore, the domain of f(x) is (-∞, -3) U [-3, 3) U [3, ∞), which simplifies to (-∞, ∞) excluding x = -3.

Range:

• For x < -3, 1/(x+3) approaches 0 from the negative side and approaches -∞ as x approaches -3 from the left.
• For -3 ≤ x < 3, x² - 9 ranges from 0 to 0.
• For x ≥ 3, √(x-3) ranges from 0 to ∞.

Therefore, combining these ranges, the range of f(x) is (-∞, 0) U [0, ∞), which simplifies to (-∞, ∞).

These examples and challenges provide a solid foundation for understanding and working with piecewise defined functions. Remember that the key lies in carefully considering the intervals and the behavior of each sub-function within its specified domain. Through consistent practice and a clear understanding of the underlying principles, you will master the complexities of piecewise functions and enhance your problem-solving abilities in mathematics. By continually engaging with these kinds of problems, you'll build a strong intuition for these versatile mathematical tools. Keep practicing, and you'll find them much less intimidating than they may initially appear.