General Functions: Behavior, Composition, and Inverses
What you must understand
General (non-analytical) functions test your ability to reason about functions from their structure rather than from a formula. Core concepts: domain and range restrictions, function notation and evaluation, composition, inverses, transformations, average rate of change, concavity, and symmetry (even/odd). You should be able to read a function's behavior from a table, a verbal description, or an algebraic expression, and to translate between representations.
Traps that show up
- Domain of a composition: the inner function's output must lie in the outer function's domain. Checking only the outer expression misses excluded inner values.
- Inverse existence: a function has an inverse only if it is one-to-one on its stated domain. A quadratic on all reals does not; restricted to x ≥ 0 it does.
- Even/odd confusion: f(−x) = f(x) is even (symmetric about the y-axis); f(−x) = −f(x) is odd (symmetric about the origin). A function can be neither.
- Average rate of change is not the derivative: it is a slope over an interval, computed as [f(b) − f(a)]/(b − a).
Rule of thumb
To find an inverse, swap x and y, solve for y, then state the domain of the inverse as the range of the original.
Worked example
Let f(x) = √(x − 2) and g(x) = x². Find the domain of f(g(x)). Compute f(g(x)) = √(x² − 2). The radicand requires x² − 2 ≥ 0, so x ≤ −√2 or x ≥ √2. Note the inner function g produces nonnegative outputs, but the outer square root still imposes its own restriction - both conditions matter.