Functions: Notation, Graphs, and Behavior
What this domain tests
Function questions ask you to interpret and connect multiple representations: equations, tables, graphs, and verbal descriptions. You must understand function notation such as $f(2)$ meaning the output when the input is $2$, and the difference between $f(x)+3$ (a vertical shift) and $f(x+3)$ (a horizontal shift left). Know domain and range, including restrictions from denominators and even roots. Be able to identify linear, quadratic, and exponential behavior from a table of values, and to describe key features of graphs: intercepts, vertex, axis of symmetry, increasing and decreasing intervals, and asymptotes. Rate of change over an interval is $\dfrac{f(b)-f(a)}{b-a}$.
Traps to watch for
Students often confuse inputs and outputs, writing $f(3)=x$ when they mean $x=3$. Transformations are a classic trap: $f(x-2)$ shifts the graph right, not left. Exponential functions grow by a constant ratio, not a constant difference, so check whether successive outputs share a multiplier. Domain questions frequently hide restrictions, such as $x\neq 2$ when a denominator is $x-2$. Also remember that a relation is a function only if each input has exactly one output.
Worked example
For $f(x)=x^2-4x+1$, find the vertex. Complete the square: $f(x)=(x^2-4x+4)-3=(x-2)^2-3$. The vertex is $(2,-3)$, and the axis of symmetry is $x=2$. Rule of thumb: for any quadratic in standard form, the vertex's x-coordinate is $-\dfrac{b}{2a}$, which here gives $-\dfrac{-4}{2}=2$.