Understand and graph piecewise-defined functions. Math 30-1 Alberta Grade 12.
Lesson 1.2 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.
A piecewise function is a function with different rules for different x-values. In mathematics, we use curly braces to write piecewise functions, with each piece having its own domain (condition).
Example. f(x) = cases x + 1 & if x < 0 2x & if x ≥ 0 cases
This means:
• When x is negative, use f(x) = x + 1
• When x is zero or positive, use f(x) = 2x
How to Read This Graph. The graph shows two rules: y = x + 1 for x < 0 and y = 2x for x ≥ 0.
Look for an open circle at (0, 1) because x < 0 does not include 0, and a closed circle at (0, 0) because x ≥ 0 includes 0.
To evaluate a piecewise function at a specific x-value, you need to first determine which piece applies, then calculate using that piece's formula.
The Process.
⚠️ Common Mistake. Students often use the wrong piece because they don't carefully check the inequalities. Always verify that your x-value satisfies the condition before using that formula!
Let's practice evaluating a simple two-piece function. We'll find values on both sides of the boundary at x = 1.
Given. f(x) = cases x + 5 & if x ≤ 1 2x & if x > 1 cases
Notice: The first piece uses ≤ (includes 1), the second uses > (doesn't include 1)
Find f(-3). Step 1: Identify which piece to use
We need to check which condition x = -3 satisfies.
• First piece condition: Is -3 ≤ 1? Yes ✓
(because -3 is less than 1)
• Second piece condition: Is -3 > 1? No ✗
(because -3 is not greater than 1)
Conclusion: Use the first piece where f(x) = x + 5
Step 2: Substitute and evaluate
Replace x with -3 in the formula f(x) = x + 5:
f(-3) = (-3) + 5
f(-3) = 2
Answer: 2
Find f(4). Step 1: Identify which piece to use
We need to check which condition x = 4 satisfies.
• First piece condition: Is 4 ≤ 1? No ✗
(because 4 is not less than or equal to 1)
• Second piece condition: Is 4 > 1? Yes ✓
(because 4 is greater than 1)
Conclusion: Use the second piece where f(x) = 2x
Step 2: Substitute and evaluate
Replace x with 4 in the formula f(x) = 2x:
f(4) = 2(4)
f(4) = 8
Answer: 8
💡 What About f(1)?. The boundary point x = 1 is where the pieces meet. Which piece should we use?
Check both conditions:
• First piece: Is 1 ≤ 1? Yes ✓ (because 1 equals 1)
• Second piece: Is 1 > 1? No ✗ (because 1 is not greater than 1)
The symbol ≤ means "less than or equal to", so when x = 1 exactly, it satisfies the first condition.
Therefore, use the first piece:
f(1) = 1 + 5 = 6
Key point: When a boundary value appears in multiple pieces, it can only belong to one piece. The inequality symbols (≤, <, >, ≥) tell you which one!
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