ACT Math: Domain and Range—Identify Restrictions at a Glance

Domain is the set of all possible input (x) values. Range is the set of all possible output (y) values. Restrictions occur when: (1) Division by zero (denominator cannot be 0). (2) Even roots of negatives (square root of -1 is undefined in reals). (3) Real-world context (time cannot be negative, distance must be positive). Example: f(x)=1/(x-2) has domain all real numbers except x=2 (would cause division by zero). Range is all real numbers except y=0 (the function approaches but never equals 0). Always scan for values that would make the function undefined, then exclude them from the domain.

From a graph: domain is the horizontal span; range is the vertical span. If the graph exists from x=-1 to x=5, domain is [-1, 5]. If y goes from -3 to 10, range is [-3, 10].

Restriction 1 (Denominators): f(x)=3/(x²-4). Denominator is 0 when x²=4 → x=±2. Domain: all reals except ±2. Written: (-∞,-2)∪(-2,2)∪(2,∞). Restriction 2 (Square roots): g(x)=√(x+5). Cannot take square root of negative. x+5≥0 → x≥-5. Domain: [-5, ∞). Restriction 3 (Real-world): Height h(t)=-16t²+100t models a ball thrown upward. Time t must be ≥0. Ball lands when h=0. Solve -16t²+100t=0 → t(100-16t)=0 → t=0 or t=6.25. Domain: [0, 6.25]. Range: [0, max height]. For each function, identify which type of restriction applies, then exclude the forbidden values.

Practice: Always ask: "Can x be anything, or are there forbidden values?" Forbidden values come from denominators, even roots, or context.

Function 1: f(x)=(x+3)/(x-1). Denominator is 0 when x=1. Domain: all reals except 1, or (-∞,1)∪(1,∞). Range: all reals except y=1 (horizontal asymptote). Function 2: g(x)=√(2x-8). 2x-8≥0 → x≥4. Domain: [4,∞). Range: [0,∞). Function 3: h(x)=x² for -2≤x≤3. Domain: [-2,3]. x² reaches minimum 0 (at x=0) and maximum 9 (at x=3). Range: [0,9]. Complete these three daily until you quickly identify restrictions and write domain/range in interval notation.

Verify by testing boundary values: if x=1 is excluded, check that f(1) is undefined.

Domain and range questions appear in 1-2 ACT Math sections, usually asking you to identify restrictions or interpret domain/range from a graph. They reward careful observation and understanding of function behavior. A student who systematically checks for restrictions answers these confidently; one who guesses misses the restrictions.

Master domain and range identification in one study session. By test day, spotting restrictions becomes automatic.