Functions
What you must understand
Functions describe relationships between inputs and outputs. You must interpret function notation, domain and range, rate of change, transformations, and inverses, and connect algebraic, graphical, and tabular representations. Linear, quadratic, polynomial, exponential, logarithmic, and trigonometric families all appear. Know how shifts, stretches, and reflections change a parent graph, and how to find an inverse by swapping variables.
Traps that appear in questions
- Domain restrictions. Square roots require nonnegative radicands; denominators cannot be zero; logarithms need positive arguments.
- Transformation order. In f(x − h) + k, the horizontal shift is opposite in sign to what students expect, and horizontal stretches act on the input, not the output.
- Confusing f(x) + 2 with f(x + 2). The first shifts up; the second shifts left.
- Inverse misconceptions. f⁻¹(x) is not 1/f(x). Only one-to-one functions have inverses that are functions.
- Average vs instantaneous rate of change. Tables give average rates over intervals.
Worked example
For f(x) = √(x − 3) + 2, the domain requires x − 3 ≥ 0, so x ≥ 3. The range is y ≥ 2 because the square root is never negative. The graph is the parent square root shifted right 3 and up 2. Rule of thumb: find domain restrictions first, then apply transformations to the parent graph.
Reading graphs
Identify intercepts, intervals of increase and decrease, and end behavior. A function's graph tells you where outputs are positive, negative, or undefined.