Functions: Definitions, Linear vs. Nonlinear, Rate of Change, and Comparisons
The Functions domain checks whether you can recognize a function from a table, graph, equation, or verbal description, distinguish linear from nonlinear functions, and compare two functions presented in different forms. Rate-of-change and initial-value questions appear most often, frequently with function notation.
Definition of a function
A function assigns exactly one output to each input. In a table, no input may appear twice with different outputs. On a graph, apply the vertical line test: if any vertical line meets the graph more than once, it is not a function.
Linear vs. nonlinear
A linear function has a constant rate of change and graphs as a straight line, writable as f(x) = mx + b. In a table with equal input steps, check whether the output differences are constant. If they change, the function is nonlinear; quadratic and exponential patterns vary in rate.
Rate of change and initial value
Rate of change = change in output ÷ change in input = Δy/Δx. The initial value is the output when the input is 0, which is b in y = mx + b and is read as the y-intercept on a graph.
Comparing functions
Convert both functions to the same representation. Slopes compare growth rates; intercepts compare starting values. A steeper line has the larger rate of change in absolute value, but a negative slope means the output falls as the input rises.
Function notation
f(3) means substitute 3 for x. If f(x) = 2x − 5, then f(3) = 1. Solving f(x) = 11 means 2x − 5 = 11, so x = 8.
Worked example
A table shows (0, 5), (2, 11), (4, 17), (6, 23). The rate of change is (11 − 5)/(2 − 0) = 3 and (17 − 11)/(4 − 2) = 3, so the function is linear with f(x) = 3x + 5. Compare with g(x) = 2x + 8. At x = 0, g starts higher (8 > 5), but f grows faster (3 > 2). The functions are equal where 3x + 5 = 2x + 8, so x = 3, and both outputs equal 14 there.
Traps
- Judging linearity from the y-column alone without confirming that the x-steps are equal.
- Computing rate of change as y/x instead of Δy/Δx.
- Reading a graph’s y-intercept as its slope.
- Accepting a graph as a function when a vertical line crosses it twice, as with a sideways parabola.
- Treating f(x) as 'f times x' rather than as the output.