This worksheet will help you master piecewise functions, a crucial concept in Algebra 2. We'll cover evaluating piecewise functions, graphing them, and writing their equations. Understanding piecewise functions is key to tackling more advanced math concepts later on.
What are Piecewise Functions?
Piecewise functions are defined by multiple sub-functions, each applying to a specific interval of the domain. Imagine it like a puzzle; each piece fits into a particular section to create the complete picture. The function's output depends entirely on which interval the input value falls within.
The general format looks like this:
f(x) = { a(x) if x < c
b(x) if c ≤ x < d
...etc...
}
Where a(x), b(x), etc., are individual functions, and c, d, etc., define the intervals.
Evaluating Piecewise Functions
Let's practice evaluating a piecewise function. Consider the following example:
f(x) = { x² if x < 2
3x - 2 if x ≥ 2
}
Problem 1: Find f(1), f(2), and f(3).
To solve this, we need to determine which sub-function to use based on the input value (x).
- f(1): Since 1 < 2, we use the first sub-function: f(1) = 1² = 1
- f(2): Since 2 ≥ 2, we use the second sub-function: f(2) = 3(2) - 2 = 4
- f(3): Since 3 ≥ 2, we use the second sub-function: f(3) = 3(3) - 2 = 7
Problem 2: Find f(-2), f(0), and f(5) for the function below:
g(x) = { |x| + 1 if x < 0
√x if x ≥ 0
}
(Solve this problem yourself. The solutions are at the end of the worksheet.)
Graphing Piecewise Functions
Graphing piecewise functions involves graphing each sub-function within its designated interval. Pay close attention to the endpoints; they might be included (closed circle) or excluded (open circle) depending on the inequality.
Problem 3: Graph the function f(x) from Problem 1.
(This requires you to graph x² for x < 2 and 3x - 2 for x ≥ 2. Remember to use an open circle at (2,4) for x² and a closed circle at (2,4) for 3x -2. This shows continuity at x = 2.)
Problem 4: Graph the function g(x) from Problem 2.
(Graph |x|+1 for x < 0 and √x for x ≥ 0. Consider the behavior near x=0.)
Writing Piecewise Function Equations
Given a graph, you can write the piecewise function's equation by identifying the individual functions and their corresponding intervals.
Problem 5: Write the piecewise function equation for the graph that has the following characteristics: A line segment from (0,2) to (3,5) and a horizontal line y = 5 for x > 3.
(Hint: Find the equation of the line segment first using the two points. The horizontal line is already defined.)
Problem 6: Write the piecewise function for a graph that includes: A parabola with vertex at (1,-1) and opening upwards for x ≤ 1, and a straight line passing through (1,0) and (3,2) for x > 1. (Hint: Remember parabola equation form, and find the equation of the line.)
Solutions to Problems 2, 4, and 5:
Problem 2: f(-2) = 3, f(0) = 0, f(5) = √5
Problem 4 & 5: These require graphical representations; it's best to sketch these out using graph paper or software. For Problem 5, you will find the equation of the line segment to be y = x + 2 for 0 ≤ x ≤ 3; combined with y = 5 for x > 3. For problem 6, you'll need to utilize your knowledge of parabola equations (vertex form) and straight line equations (point-slope or slope-intercept form).
This worksheet provides a solid foundation for understanding and working with piecewise functions. Remember to practice more problems to solidify your understanding. Good luck!
