piecewise functions worksheet answer key

Lemon

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31-01-2025

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Piecewise Functions Worksheet: Answer Key & Explanations

This answer key provides solutions and detailed explanations for a typical piecewise function worksheet. Since I don't have access to your specific worksheet, I'll cover common problem types encountered when working with piecewise functions. Remember to always refer to your specific worksheet's instructions and notations.

What are Piecewise Functions?

Piecewise functions are defined by multiple sub-functions, each applicable over a specific interval of the domain. They are often represented graphically as a collection of separate function segments. Understanding the domain restrictions is crucial for evaluating and graphing these functions.

Common Problem Types & Solutions:

1. Evaluating Piecewise Functions:

This involves substituting a given x-value into the correct sub-function based on the defined intervals.

Example:

Let's say we have the piecewise function:

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

a) Find f(-2):

Since -2 < 0, we use the first sub-function: f(-2) = 2(-2) + 1 = -3

b) Find f(0):

Since 0 ≥ 0, we use the second sub-function: f(0) = (0)² - 3 = -3

c) Find f(3):

Since 3 ≥ 0, we use the second sub-function: f(3) = (3)² - 3 = 6

2. Graphing Piecewise Functions:

Graphing requires plotting the sub-functions within their specified intervals. Pay close attention to endpoints – whether they are included (closed circle) or excluded (open circle).

Example (using the same f(x) as above):

To graph f(x), you would:

• Graph y = 2x + 1 for all x-values less than 0 (open circle at x=0).
• Graph y = x² - 3 for all x-values greater than or equal to 0 (closed circle at x=0).

3. Finding the Domain and Range:

The domain is the set of all possible x-values, and the range is the set of all possible y-values.

Example (using the same f(x) as above):

• Domain: The domain of f(x) is all real numbers, (-∞, ∞), because there are no restrictions on the x-values across the defined intervals.
• Range: To find the range, analyze the graph. For the first part (2x+1), y values are less than 1 (excluding 1). For the second part (x²-3), y values are greater than or equal to -3. Therefore, the range is (-∞, 1) U [-3, ∞).

4. Solving Equations Involving Piecewise Functions:

This involves determining which sub-function to use based on the potential solution and solving the resulting equation.

Example:

Solve f(x) = 5, using the same f(x) as above.

We need to consider both sub-functions:

• 2x + 1 = 5: This gives x = 2. However, this solution is not valid because the first sub-function is only defined for x < 0.
• x² - 3 = 5: This gives x² = 8, so x = ±√8 = ±2√2. Since both 2√2 and -2√2 are ≥ 0, both are valid solutions.

5. Determining Continuity:

A piecewise function is continuous at a point if the function value at that point matches the limit of the function as x approaches that point from both sides. Discontinuities are often 'jumps' or 'holes' in the graph. Checking continuity at the boundaries between sub-functions is key.

General Tips for Solving Piecewise Function Problems:

• Carefully examine the definitions of the sub-functions and their corresponding intervals. This is the most critical step.
• Visualize the graph. Sketching a rough graph can greatly aid in understanding the function's behavior and solving problems.
• Pay close attention to endpoints. Determine whether the endpoints are included or excluded in each interval.
• Check your work. Substitute your answers back into the original function to verify their correctness.

This comprehensive guide should help you solve most problems on a typical piecewise function worksheet. Remember to adapt these explanations to your specific worksheet's questions. If you have a specific problem you're stuck on, feel free to provide the function and question, and I'll help you work through it!