Functions
Functions
The Functions domain tests function notation, domain and range, rates of change, linear and exponential models, transformations, and interpretation of graphs. Expect items that ask you to compare two functions represented differently-equation, table, graph, or verbal description.
Core rules
- f(a) means substitute a for the input; f(x) = k means solve for the input that produces output k.
- A linear function has a constant rate of change m = (y2 - y1)/(x2 - x1); an exponential function has a constant ratio b and form f(x) = a·b^x.
- Domain restrictions come from context (time cannot be negative) and from algebra (denominators cannot be zero).
- Transformations: f(x) + k shifts up k; f(x - h) shifts right h; a·f(x) stretches vertically by |a| and reflects across the x-axis if a < 0.
- Key features to identify: intercepts, intervals of increase and decrease, maximum or minimum, and end behavior.
Worked example
Let f(x) = 3x - 5 and g(x) = 2·3^x. Find and compare f(4) and g(4). f(4) = 3(4) - 5 = 7. g(4) = 2·3^4 = 2·81 = 162. Both increase, but f grows by a constant addend of 3 per unit input, while g multiplies by 3 each step. Over equal intervals, the exponential function eventually overtakes the linear one.
Traps
Reading f(x + 2) as f(x) + 2; assuming every graph that rises is linear; ignoring domain limits in context; and confusing the base of an exponential function with its initial value.