graphing piecewise functions worksheet with answers pdf

Olvin

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12-01-2025

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This worksheet provides a thorough guide to graphing piecewise functions, a crucial concept in algebra and precalculus. We'll cover the fundamental steps involved, illustrate them with clear examples, and provide solutions so you can check your understanding. Mastering piecewise functions is key to understanding more advanced mathematical concepts.

Understanding Piecewise Functions

A piecewise function is a function defined by multiple subfunctions, each applicable over a specific interval of the domain. It's like a collection of different functions stitched together to create a single, often discontinuous, function. The key to graphing them lies in understanding the individual subfunctions and their respective domains.

The general form of a piecewise function is:

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

Where g(x), h(x), and k(x) are different functions, and a, b, and c represent the boundaries of their respective intervals.

Key Steps to Graphing Piecewise Functions

1. Identify the Subfunctions and their Intervals: Carefully examine the definition of the piecewise function. Note each subfunction and the specific interval (or domain) where it applies.

2. Graph Each Subfunction Individually: For each subfunction, graph it as you would any other function. Pay close attention to the boundaries of the interval.

3. Restrict the Graph to the Specified Interval: This is crucial. Only keep the portion of each subfunction's graph that falls within its designated interval. Erase any part of the graph that lies outside that interval.

4. Combine the Graphs: Bring together the restricted graphs of each subfunction. The resulting combined graph represents the piecewise function. Points at the boundaries of the intervals need special attention, ensuring continuity or correctly indicating discontinuity.

5. Check for Open and Closed Circles: Use open circles (◦) to indicate points excluded from the graph (e.g., x < b), and closed circles (•) to indicate points included (e.g., x ≥ b). This visually represents the function's behavior at the boundaries of the intervals.

Worked Examples

Let's work through a couple of examples to solidify our understanding.

Example 1:

Graph the following piecewise function:

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

Solution:

1. Subfunctions and Intervals: We have two subfunctions: f(x) = x + 2 (for x < 1) and f(x) = x² (for x ≥ 1).

2. Individual Graphs: Graph y = x + 2 and y = x² separately.

3. Restriction: For y = x + 2, only keep the portion to the left of x = 1 (open circle at x = 1). For y = x², keep the entire graph from x = 1 onwards (closed circle at x = 1).

4. Combination: Combine the restricted graphs. You'll have a straight line segment ending at (1, 3) with an open circle, connected to a parabola starting at (1, 1) with a closed circle.

Example 2:

Graph the piecewise function:

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

Solution:

Follow the same four steps as above. You'll end up with a horizontal line at y = -1 for x ≤ -2 (closed circle at x = -2), a line segment from (-2, -1) to (1, 2) (open circles at both ends), and a horizontal line at y = 2 for x ≥ 1 (closed circle at x = 1).

Practice Problems

Now it's your turn! Try graphing these piecewise functions and check your answers against the solutions provided at the end (solutions are not included here as it would defeat the purpose of the worksheet, requiring self-working and checking). Remember to follow the steps outlined above.

(Practice problems would be included here. These would involve various piecewise functions with increasing complexity to test comprehension.)

This worksheet provides a foundation for understanding and graphing piecewise functions. Practice is key to mastering this important skill. Remember to always carefully consider the intervals and the behavior of the function at the boundaries.