ACT Math: Piecewise Functions—Evaluate, Graph, and Find Domain and Range

A piecewise function is defined by different rules on different intervals. Example: f(x)={2x if x<1; x^2 if x≥1}. For x=0.5: f(0.5)=2(0.5)=1. For x=2: f(2)=2^2=4. Each piece applies only in its domain interval. On the ACT, you'll: (1) Evaluate piecewise functions at specific values, (2) Graph them (each piece is a line, curve, etc.; transitions matter), (3) Find domain and range, (4) Identify discontinuities (jumps or breaks). Graphing: Each piece is a function on its interval. The transition point (boundary) determines whether the function is continuous. If the left limit equals the right limit at the boundary, the function is continuous; otherwise, it's discontinuous (jump). Piecewise functions model real situations: phone plans (different rates for different minutes), tax brackets (different rates for different income levels), parking fees (flat rate, then per hour).

Key: Always check which interval x belongs to before evaluating. Using the wrong formula is a common error.

Mistake 1: Using the wrong rule. f(x)={2x if x<1; x^2 if x≥1}. For x=1, you need x^2 (because 1≥1), not 2x. f(1)=1, not 2. Mistake 2: Forgetting to check boundaries. Is f(1)=1 or another value? Check if 1 is included in which interval. Mistake 3: Graphing without attention to endpoints. A piece might be an open circle (not included, x<1) or closed circle (included, x≥1). Mistake 4: Assuming piecewise functions are always continuous. Many have jumps. Identify continuity explicitly. At boundaries, always verify which interval applies and whether the endpoint is included.

Checklist: (1) For evaluation, identify which interval x is in. (2) Use the corresponding rule. (3) When graphing, plot each piece, marking endpoints carefully (open vs. closed). (4) Identify discontinuities (jumps) if they exist. (5) Find domain (typically all x for which a rule applies) and range (all y-values the function produces).

Function 1: f(x)={x+1 if x≤0; 2x if 0For each function, identify intervals, plot each piece, mark endpoints, and explain continuity or discontinuity at boundaries.

Daily drill: Solve one piecewise function problem daily. Alternate between evaluation and graphing. Practice identifying discontinuities.

About 1 piecewise function question per ACT Math section combines domain, function evaluation, and graphing—multiple skills at once. These feel advanced but follow straightforward logic: identify the right rule for the input, apply it, and observe the result. Correctly solving a piecewise function problem signals comfort with function notation and piecewise reasoning, making these high-value questions where right answers demonstrate deeper understanding.

Spend 2 days on piecewise functions. Practice evaluating at various x-values, graphing with attention to endpoints, and identifying discontinuities. By test day, you'll handle piecewise functions confidently.