5-2 Additional Practice Piecewise Defined Functions

5+2 Additional Practice Piecewise Defined Functions: Mastering the Art of Defining Functions in Pieces

Piecewise defined functions, those mathematical chameleons that change their behavior depending on the input, can seem daunting at first. However, with consistent practice and a solid understanding of their core principles, you'll find them surprisingly manageable and even elegant. This article provides five core examples of piecewise defined functions, followed by two bonus challenges to solidify your understanding. We'll explore how to graph them, evaluate them, and even apply them to real-world scenarios.

Understanding the Foundation: What are Piecewise Defined Functions?

A piecewise defined function is simply a function defined by multiple sub-functions, each applicable over a specified interval of its domain. Think of it as a function with multiple personalities, each taking the stage depending on the input value. The crucial aspect is defining the domain for each sub-function, ensuring there's no overlap or gaps in the overall function's domain. This is typically represented using a combination of equations and conditions, often enclosed within curly braces {}.

Example 1: The Absolute Value Function – A Classic Piecewise Representation

The absolute value function, denoted as |x|, is a prime example of a piecewise defined function. Its definition is elegantly simple:

|x| = { x, if x ≥ 0 {-x, if x < 0

This means that for non-negative values of x, the function returns x itself. However, for negative values, the function returns the negation of x, effectively making it positive. This ensures the output is always non-negative.

Graphing this function reveals a V-shape, with the vertex at the origin (0,0). The left branch has a slope of -1, while the right branch has a slope of 1.

Evaluating this function:

• |3| = 3
• |-5| = 5
• |0| = 0

Example 2: The Greatest Integer Function (Floor Function)

The greatest integer function, often denoted as ⌊x⌋ or floor(x), returns the greatest integer less than or equal to x. This function is inherently piecewise because it "jumps" at each integer value.

⌊x⌋ = { ...,-2, if -2 ≤ x < -1 {-1, if -1 ≤ x < 0 { 0, if 0 ≤ x < 1 { 1, if 1 ≤ x < 2 { 2, if 2 ≤ x < 3 { ...

While we can't list all intervals, the pattern is clear. The function always rounds down to the nearest integer.

Graphing this function shows a series of horizontal steps, each of length 1, with discontinuities at integer values.

Evaluating this function:

• ⌊3.7⌋ = 3
• ⌊-2.3⌋ = -3
• ⌊0⌋ = 0

Example 3: A Piecewise Linear Function

Let's construct a piecewise linear function:

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

This function consists of two linear segments. For x values less than 1, the function follows the line 2x + 1. For x values greater than or equal to 1, it follows the line -x + 4.

Graphing this function shows two lines meeting at the point (1,3). Note that the function is continuous at x = 1, meaning there's no gap or jump in the graph at that point.

Evaluating this function:

• f(0) = 2(0) + 1 = 1
• f(1) = -(1) + 4 = 3
• f(3) = -(3) + 4 = 1

Example 4: A Piecewise Quadratic Function

Piecewise functions aren't limited to linear segments. Let's consider a piecewise quadratic function:

g(x) = { x², if x ≤ 2 { 4x - 4, if x > 2

This function utilizes a parabola (x²) for x values less than or equal to 2 and a straight line (4x - 4) for x values greater than 2.

Graphing this function combines a portion of a parabola with a line segment. The point (2,4) is where the parabola and line meet, resulting in a continuous function.

Evaluating this function:

• g(0) = 0² = 0
• g(2) = 2² = 4
• g(4) = 4(4) - 4 = 12

Example 5: A Function with a Discontinuity

Not all piecewise functions are continuous. Let's create one with a jump discontinuity:

h(x) = { 1/(x-1), if x ≠ 1 { 2, if x = 1

This function is undefined at x = 1. Notice how the value of h(x) approaches positive or negative infinity as x approaches 1. However, the function is explicitly defined at x = 1 as 2. This creates a visible gap or jump in the graph.

Graphing this function shows a hyperbola (1/(x-1)) with a single point at (1,2) which is not on the hyperbola.

Evaluating this function:

• h(0) = 1/(-1) = -1
• h(1) = 2
• h(2) = 1/(2-1) = 1

Bonus Challenge 1: A More Complex Piecewise Function

Let's tackle a more complex piecewise function:

k(x) = { |x - 2|, if x < 3 { √(x - 3), if x ≥ 3

This function combines an absolute value function and a square root function. Analyze how the absolute value portion behaves, remember its V-shape centered around x = 2. Then consider the square root function starting at x = 3.

Graphing this function requires careful consideration of both sub-functions and their domains. At x = 3, there's a continuous transition.

Evaluating this function:

• k(0) = |0 - 2| = 2
• k(3) = √(3 - 3) = 0
• k(7) = √(7 - 3) = 2

Bonus Challenge 2: Modeling a Real-World Scenario

Imagine a taxi fare system: $2.00 for the first mile and $1.50 for each additional mile. Represent this as a piecewise function, where 'x' represents the miles driven and 'f(x)' represents the total fare.

This will involve a linear increase starting after the first mile.

The solution would be something like:

f(x) = { 2.00, if x ≤ 1 { 2.00 + 1.50(x - 1), if x > 1

This function accurately reflects the cost structure of the taxi fare.

Conclusion: Mastering Piecewise Functions

Piecewise defined functions, while initially appearing complex, become much more accessible with practice. By understanding how to break down the function into its component parts, considering the domain of each piece, and practicing evaluation and graphing, you'll gain confidence in working with these versatile mathematical tools. Remember that piecewise functions are not just abstract concepts; they frequently model real-world phenomena, making them relevant and practical in various fields. Keep practicing, and you'll be well on your way to mastering the art of piecewise functions!