Functions
Core ideas
A function assigns exactly one output to each input. Use function notation carefully: f(3) means substitute 3 for x, not multiply f by 3. Know how to identify domain and range from equations, tables, and graphs; the domain is the set of allowed inputs and the range is the set of resulting outputs. A graph represents a function if every vertical line crosses it at most once. Linear functions have a constant rate of change and graph as straight lines; nonlinear functions do not. For y = mx + b, m is the slope (change in y per change in x) and b is the y-intercept. Average rate of change over an interval is the change in f(x) divided by the change in x.
Traps
Do not assume f(a+b) = f(a)+f(b); almost no function works that way. Distinguish slope from intercept: a graph can have a large y-intercept and a small slope. When reading intervals of increase or decrease, report x-values, not y-values. A table can represent a function even if two inputs give the same output, but not if one input gives two outputs. Watch for restricted domains, such as values that make a denominator zero.
Worked example
For f(x) = 2x^2 − 3, find f(−2). Substitute: 2(−2)^2 − 3 = 2(4) − 3 = 8 − 3 = 5. Rule of thumb: substitute the input inside parentheses first, then follow order of operations; square before multiplying.