Functions: Concept, Properties, and Operations
The Functions: Concept, Properties, and Operations domain asks you to treat a function as an object: identify its domain and range, combine and compose functions, invert them, and describe symmetry and monotonicity. Expect questions that hinge on a single restriction - a denominator, a radicand, a restricted domain, or the order of operations inside a composition.
Definitions and rules
- A function assigns exactly one output to each input; the vertical line test reads this off a graph.
- Domain exclusions: denominators cannot be zero, even roots require a nonnegative radicand, and log arguments must be positive. The range is the set of outputs actually attained.
- Arithmetic combinations: (f + g)(x) = f(x) + g(x), with domain the intersection of the two domains.
- Composition (f ∘ g)(x) = f(g(x)) requires x in the domain of g and g(x) in the domain of f.
- One-to-one functions have inverses. Find f-inverse by solving y = f(x) for x, then swapping; its graph reflects f across y = x, and its domain equals the range of f.
- Even: f(-x) = f(x), symmetric about the y-axis. Odd: f(-x) = -f(x), symmetric about the origin.
- Average rate of change on [a, b] is (f(b) - f(a))/(b - a); the difference quotient is (f(x + h) - f(x))/h with h not zero.
Worked example
Let f(x) = x^2 - 4x + 7 with domain x >= 2. Complete the square: f(x) = (x - 2)^2 + 3. On x >= 2 the graph is increasing, so f is one-to-one and invertible. Write y - 3 = (x - 2)^2. Because x >= 2, the quantity x - 2 is nonnegative, so x = 2 + sqrt(y - 3). Swapping variables gives f-inverse(x) = 2 + sqrt(x - 3) with domain x >= 3.
Traps that cost marks
- Writing ±sqrt and keeping both branches after a domain restriction has already selected one.
- Assuming f ∘ g equals g ∘ f; composition is not commutative.
- Ignoring the inner function's domain restriction when composing.
- Cancelling h in the difference quotient without stating h is nonzero.
- Calling a function even or odd without testing f(-x) symbolically - many functions are neither.