Questions: Piecewise Functions — Graphing and Analysis

Question 1

For f(x) = { x², if x < 2; 3x − 1, if x ≥ 2 }, what is f(2)?

• A. 4, by substituting into x²
• B. 5, by substituting into 3x − 1
• C. 4.5, by averaging the two formulas
• D. Undefined, because 2 is a boundary point

Answer: (B)

Since x = 2 satisfies x ≥ 2, the second formula applies: 3(2) − 1 = 5. The first formula only applies when x < 2 — the strict inequality means x = 2 is excluded from that piece. Boundary points are not special exceptions; you simply determine which piece's domain they belong to.

Question 2

Consider f(x) = { x + 1, if x < 2; x + 3, if x ≥ 2 }. Is f continuous at x = 2?

• A. Yes — f(2) is defined and equals 5, so the function is continuous there
• B. Yes — piecewise functions are always continuous at their boundaries
• C. No — the left-hand limit as x → 2⁻ equals 3, but f(2) = 5, so there is a jump
• D. No — piecewise functions are never continuous at their boundaries

Answer: (C)

The left-hand limit is lim(x→2⁻) (x+1) = 3. The right-hand limit is lim(x→2⁺) (x+3) = 5 = f(2). Because the left-hand limit (3) does not equal f(2) (5), there is a jump discontinuity. Option A is the classic error: f(2) being defined is necessary but not sufficient for continuity — the limits must also agree with f(2).

Question 3

A piecewise function is generally discontinuous at its boundary points.

• T. True
• F. False

Answer: False

Piecewise functions can be perfectly continuous at their boundaries if the pieces connect without a gap or jump. For example, f(x) = { x, if x < 1; 1, if x ≥ 1 } is continuous at x = 1 because both pieces approach 1 from their respective sides. The pieces meeting at a boundary just needs left limit = right limit = f(boundary).

Question 4

For f(x) = { x² if x < 0; −x if x ≥ 0 }, f(0) = 0.

• T. True
• F. False

Answer: True

Since x = 0 satisfies x ≥ 0, we apply the second formula: f(0) = −(0) = 0. The first formula (x²) applies only for strictly negative x, so it is not used here. Note that both pieces happen to approach 0 as x → 0, making this function continuous at the boundary.

Question 5

Why must you pay careful attention to open versus closed endpoints when graphing a piecewise function?

Model answer: Whether an endpoint is open or closed determines which formula gives the function's value at that exact boundary point, and it controls whether the graph shows a filled circle (value included) or an open circle (value excluded). If x = a appears as x < a in one piece and x ≥ a in another, then a belongs to the second piece. Getting this wrong produces both an incorrect function value and a misleading graph.

Open and closed endpoints also preview the concept of continuity: if the two pieces yield the same value at a boundary, the dot and open circle coincide and the function is continuous there. If they yield different values, a jump appears. This distinction is precisely what one-sided limits formalize.