Definition
Linear function
A linear function is a function with a constant rate of change whose graph lies on a straight line.
What it means
Take equal steps in the input. If the output changes by the same signed amount every time, the function is linear. On a graph, its points line up exactly. A discrete situation may show only separate points, even though those points lie on one line.
Remember it
Equal input steps make equal output steps.
Example
For f(x) = 3x − 2, inputs 1, 2, and 3 give outputs 1, 4, and 7. Each input step of 1 adds 3 to the output, so the rate is constant and the function is linear.
Watch for
For f(x) = x², outputs 1, 4, and 9 change by 3 and then 5, so the rate is not constant and the function is not linear.
Common mix-up
A discrete linear function may appear as separate collinear points; the missing points are not automatically allowed inputs.
Exact meaning
A linear function is a function with a constant rate of change. For numerical variables over a suitable domain it can be written f(x) = mx + b; its plotted points lie on one straight line, forming the line when the domain is continuous.