Functions and graphs: domain, range, monotonicity, extrema, transformations
Read the function before computing
For every function ask four questions in order: domain, range, monotonicity, extrema. Domain restrictions come from denominators (≠ 0), even roots (radicand ≥ 0), logarithms (argument > 0) and tan (cos ≠ 0). The range is the set of outputs; for a quadratic y = a(x − h)² + k it is [k, +∞) when a > 0 and (−∞, k] when a < 0.
Monotonicity: linear functions are monotone everywhere; a quadratic decreases then increases (or the reverse) about its vertex; k/x is decreasing on each branch; aˣ is increasing for a > 1 and decreasing for 0 < a < 1. Extrema on a closed interval occur at critical points or at endpoints - check both.
Transformations: y = f(x − a) shifts right by a; y = f(x) + b shifts up by b; y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis; y = k·f(x) stretches vertically by k. Order matters: handle horizontal shifts and stretches inside the bracket first, then vertical changes outside.
Traps. Applying transformations in the wrong order. Assuming the range of f(g(x)) equals the range of f. Forgetting that an even function satisfies f(−x) = f(x), an odd one f(−x) = −f(x), and that many functions are neither. Confusing the domain of f⁻¹ with that of f; they swap.
Worked example. y = 2(x − 3)² − 5 has vertex (3, −5), opens upward, minimum value −5 at x = 3, range [−5, +∞), decreasing on (−∞, 3] and increasing on [3, +∞).