This answer key provides solutions to common piecewise function problems, along with explanations to solidify your understanding. Piecewise functions, while initially appearing complex, are simply functions defined by different sub-functions across distinct intervals of their domain. Mastering them is crucial for various mathematical applications.
What are Piecewise Functions?
A piecewise function is a function defined by multiple sub-functions, each applicable over a specified interval of the domain. These intervals are often disjoint (non-overlapping), creating a function with distinct behavior across different parts of its domain. The key is to correctly identify which sub-function to use based on the input value.
Example Problems & Solutions:
Let's tackle some typical piecewise function problems, providing detailed solutions for clarity.
Problem 1:
Evaluate the piecewise function f(x) defined as:
f(x) = { 2x + 1, if x < 0
{ x² - 3, if x ≥ 0
Find: a) f(-2) b) f(0) c) f(3)
Solution 1:
a) Since -2 < 0, we use the first sub-function: f(-2) = 2(-2) + 1 = -3
b) Since 0 ≥ 0, we use the second sub-function: f(0) = (0)² - 3 = -3
c) Since 3 ≥ 0, we use the second sub-function: f(3) = (3)² - 3 = 6
Problem 2:
Graph the following piecewise function:
g(x) = { |x|, if x < 2
{ 4, if x ≥ 2
Solution 2:
To graph this function, consider each sub-function separately:
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For x < 2: The function is g(x) = |x|. This is the absolute value function, creating a V-shape with the vertex at (0,0). However, we only graph the portion where x < 2. There will be an open circle at (2,2) because x does not equal 2 in this interval.
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For x ≥ 2: The function is g(x) = 4. This is a horizontal line at y = 4. The graph will be a solid line starting from x = 2 extending infinitely to the right.
(Graphical representation would be included here if this were a visual document. Describe the graph to help the user visualize) The graph will show a V-shape from negative infinity to (2,2) (open circle) and a horizontal line at y=4 starting from (2,4) (closed circle) extending infinitely to the right.
Problem 3:
Write a piecewise function representing the following scenario: A taxi charges $3 for the first mile and $2 for each additional mile.
Solution 3:
Let C(m) be the cost for m miles.
C(m) = { 3, if 0 < m ≤ 1
{ 3 + 2(m-1), if m > 1
This function reflects the $3 initial charge and the subsequent $2 per additional mile.
Advanced Concepts & Further Practice:
Once you've grasped the basics, challenge yourself with these more advanced topics:
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Continuity and Discontinuity: Analyze where a piecewise function is continuous and where it has discontinuities (jumps or breaks in the graph).
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Piecewise Linear Functions: Focus on functions composed entirely of linear segments.
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Domain and Range: Determine the domain (all possible x-values) and range (all possible y-values) of piecewise functions.
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Solving Equations with Piecewise Functions: Practice solving equations where the unknown is within the piecewise function definition.
By working through various examples and tackling more complex problems, you'll build a solid foundation in understanding and applying piecewise functions. Remember to carefully consider the intervals and the corresponding sub-functions when evaluating or graphing these unique functions.
