Functions and Relations: Definitions, Properties, Equations and Graphs
What you must understand
A function assigns each input exactly one output. You should be able to determine a function's domain and range, read properties from a formula or a graph (monotonicity, extrema, parity, injectivity), and work with the standard families: linear, quadratic, power, exponential, logarithmic, and trigonometric. You must also solve equations and inequalities involving these functions and understand transformations of graphs.
Traps that appear in questions
- Domain of composite or root/log functions. For (f(x)=\log(x-2)), the domain is (x>2), not (x\ge 2). For (\sqrt{x-3}), it is (x\ge 3).
- Parity confusion. Even functions satisfy (f(-x)=f(x)) (symmetric about the y-axis); odd functions satisfy (f(-x)=-f(x)) (symmetric about the origin). Neither is not the same as odd.
- Quadratic vertex. The vertex of (y=ax^2+bx+c) is at (x=-b/(2a)); the sign of (a) tells whether it is a maximum or minimum.
- Exponential vs power. (2^x) grows differently from (x^2); do not mix the laws.
- Transformations. (f(x-h)) shifts right by (h) (counter-intuitive sign), and (-f(x)) reflects across the x-axis.
Worked example
Find the vertex and range of (f(x)=x^2-4x+1). Vertex x-coordinate: (-b/(2a)=4/2=2). Value: (f(2)=4-8+1=-3). Since (a=1>0), the parabola opens upward, so the minimum is (-3) and the range is (\langle -3, \infty)).
Rule of thumb: before analysing any function, write down its domain first - many exam errors come from evaluating a function at a point where it is not defined.