evaluating piecewise functions answer key

Lemon

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01-02-2025

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Evaluating piecewise functions can seem daunting at first, but with a systematic approach and a solid understanding of the underlying concepts, it becomes straightforward. This guide serves as an answer key for common piecewise function evaluation problems, along with a comprehensive explanation to solidify your understanding. We'll cover various scenarios, from simple evaluations to more complex examples involving inequalities and absolute values.

Understanding Piecewise Functions

A piecewise function is a function defined by multiple sub-functions, each applicable over a specified interval of the domain. The key is identifying which sub-function to use based on the input value (x). The function is defined as a collection of pieces, hence the name "piecewise."

General Form:

A piecewise function is typically written in the following form:

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 are the boundaries defining the intervals.

Evaluating Piecewise Functions: Step-by-Step

Here's a step-by-step process to evaluate piecewise functions:

1. Identify the Input: Determine the value of 'x' you need to evaluate the function for.

2. Determine the Correct Sub-function: Examine the intervals defined in the piecewise function. Find the interval that contains your input value 'x'.

3. Substitute and Evaluate: Substitute the input value 'x' into the corresponding sub-function and perform the necessary calculations.

Example Problems and Solutions

Let's work through some examples to illustrate the process.

Example 1:

Evaluate the function:

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

for the following values:

a) f(1) b) f(2) c) f(3)

Solutions:

a) f(1): Since 1 < 2, we use the first sub-function: f(1) = 2(1) + 1 = 3

b) f(2): Since 2 ≥ 2, we use the second sub-function: f(2) = (2)² - 1 = 3

c) f(3): Since 3 ≥ 2, we use the second sub-function: f(3) = (3)² - 1 = 8

Example 2 (Involving Absolute Value):

Evaluate the function:

g(x) = {
  |x| - 2, if x ≤ 0
  x - 2, if x > 0
}

for the following values:

a) g(-3) b) g(0) c) g(5)

Solutions:

a) g(-3): Since -3 ≤ 0, we use the first sub-function: g(-3) = |-3| - 2 = 1

b) g(0): Since 0 ≤ 0, we use the first sub-function: g(0) = |0| - 2 = -2

c) g(5): Since 5 > 0, we use the second sub-function: g(5) = 5 - 2 = 3

Example 3 (More Complex Intervals):

Evaluate:

h(x) = {
  x + 1, if -1 ≤ x < 0
  x², if 0 ≤ x < 2
  3x - 2, if x ≥ 2
}

for x = -0.5, x = 1, and x = 3.

Solutions:

• h(-0.5): Since -1 ≤ -0.5 < 0, h(-0.5) = -0.5 + 1 = 0.5
• h(1): Since 0 ≤ 1 < 2, h(1) = 1² = 1
• h(3): Since 3 ≥ 2, h(3) = 3(3) - 2 = 7

Tips for Success

• Pay close attention to the inequalities defining the intervals. These inequalities determine which sub-function to use.
• Always double-check your work. A small error in substitution can lead to an incorrect answer.
• Practice, practice, practice! The more examples you work through, the more confident you'll become in evaluating piecewise functions.

This guide provides a solid foundation for evaluating piecewise functions. Remember to carefully analyze the intervals and correctly substitute the input value into the appropriate sub-function. With consistent practice, you'll master this essential mathematical skill.