Functions
Functions describe a relationship in which each input has exactly one output. This domain tests whether you can identify functions from tables, graphs, equations, and verbal rules; compare their properties; and model linear and nonlinear relationships.
Defining and representing functions
A relation is a function if every input value is paired with exactly one output value. In a table, repeated inputs with different outputs mean “not a function.” On a graph, use the vertical line test: if any vertical line crosses the graph more than once, it is not a function. Function notation f(x) = 3x + 2 means the output is 3 times the input plus 2, so f(4) = 14.
Comparing functions
Compare functions using rate of change and initial value, even when one is an equation, one is a table, and one is a graph. For a linear function y = mx + b, the rate of change is m and the initial value is b. From a table, rate of change = change in output ÷ change in input over equal intervals; from a graph, use two clear points. A greater rate of change means a steeper line; for decreasing lines, a more negative rate is steeper.
Linear versus nonlinear
A linear function has a constant rate of change, and its graph is a straight line. A nonlinear function does not. In a table, check whether output differences are constant for equal input steps; if not, the function is nonlinear. Equations with x², |x|, or x in a denominator are generally nonlinear. Graphs of nonlinear functions curve.
Modeling
Translate a verbal situation into a function by identifying the starting amount and the repeated change. If a tank starts with 50 gallons and drains 4 gallons per minute, the amount after t minutes is A(t) = 50 − 4t. State the domain that makes sense in context, such as 0 ≤ t ≤ 12.5.
Worked example
Function A is y = 4x − 1. Function B is given by the table (0, 3), (2, 11), (4, 19). For B, rate = (11 − 3)/(2 − 0) = 4, and initial value = 3. Both have rate 4, but B starts at 3 and A starts at −1, so B is always 4 greater. At x = 5, A = 19 and B = 23.
Traps that cost marks
- Calling a relation a function when an input repeats with different outputs.
- Comparing only y-intercepts or only slopes when the question asks for a full comparison.
- Assuming a table is linear without checking equal input intervals.
- Reading the initial value from a graph incorrectly when the x-axis does not start at 0.
- Ignoring context restrictions on the domain.