Functions and Their Graphs
Functions and Their Graphs
This domain tests function concepts rather than isolated equation solving: identifying functions, evaluating function notation, describing domain and range, reading key features from a graph, and applying transformations. Expect items that give a graph or table and ask for intercepts, intervals of increase or decrease, or average rate of change.
Core rules
- A relation is a function when each input has exactly one output (it passes the vertical line test; no repeated x-value pairs with different y-values).
- f(a) means substitute a for x. f(x + 2) shifts the graph 2 units left; f(x) + 2 shifts it 2 units up.
- Transformations: a·f(x) stretches vertically by |a| and reflects over the x-axis when a < 0; f(bx) compresses horizontally by 1/|b|; −f(x) reflects over the x-axis; f(−x) reflects over the y-axis.
- Domain exclusions: set denominators ≠ 0 and require even-root radicands to be ≥ 0.
- Average rate of change on [a, b] = (f(b) − f(a))/(b − a), the slope of the secant line.
- The inverse is found by swapping x and y and solving; its graph reflects over the line y = x.
Worked example
For f(x) = x² − 4x + 1, find the average rate of change on [0, 2].
f(0) = 1 and f(2) = 4 − 8 + 1 = −3. Average rate of change = (−3 − 1)/(2 − 0) = −4/2 = −2. Over that interval the function decreases by 2 units per unit of x on average.
Traps that cost marks
- Reversing a horizontal shift: f(x − 3) moves the graph right, not left.
- Reporting f(b) − f(a) and omitting the division by (b − a).
- Confusing domain (inputs, x-values) with range (outputs, y-values).
- Calling a curve a function because it looks smooth, when it fails the vertical line test.