Evaluating Piecewise Functions
Evaluating piecewise functions requires identifying which domain interval contains the input value, then applying only that piece's rule — a core skill in enVision Algebra 1 Chapter 5 for Grade 11. Given f(x) = {2x+1 if x<3, x² if x≥3}, finding f(2) uses the first piece because 2 < 3, giving 2(2)+1 = 5. Finding f(3) uses the second piece because 3 ≥ 3, giving 3² = 9. The critical step is checking inequality signs carefully, especially at boundary values, to determine which rule applies. Never apply both pieces to the same input.
Key Concepts
To evaluate a piecewise function at a given input value, determine which domain interval contains the input, then apply the corresponding rule for that piece.
Common Questions
What is the key first step when evaluating a piecewise function?
Determine which domain interval contains your input value by checking each condition. Once you find the matching interval, apply only that piece's formula.
For f(x) = {2x+1 if x<3, x² if x≥3}, what is f(3)?
Since 3 ≥ 3, use the second piece: f(3) = 3² = 9.
For the same function, what is f(-1)?
Since -1 < 3, use the first piece: f(-1) = 2(-1) + 1 = -1.
How do inequality signs affect which piece to use at a boundary?
A strict inequality (<) means the boundary value belongs to the other piece. In the example, x = 3 satisfies x ≥ 3 but not x < 3, so x = 3 uses the second piece.
Can the same x-value use two different pieces of a piecewise function?
No. A well-defined piecewise function assigns each x-value to exactly one piece. The domain intervals are non-overlapping, so each input maps to exactly one output.