In mathematics, the term linear function can refer to either of two different but related concepts.
Usage in elementary mathematics
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In elementary algebra and analytic geometry, the term linear function is sometimes used to mean a first degree polynomial function of one variable. These functions are called "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.
Such a function can be written as
- f(x) = mx + b
(called slope-intercept form), where m and b are real constants and x is a real variable. The constant m is often called the slope while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.
Three geometric linear functions — the red and blue ones have the same slope ( m), while the red and green ones have the same y-intercept ( b).
Examples of functions whose graph is a line include the following:
- f1(x) = 2x + 1
- f2(x) = x / 2 + 1
- f3(x) = x / 2 − 1
The graphs of these are shown in the image at right.
Usage in advanced mathematics
In advanced mathematics, a linear function often means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.
For example, if x and f(x) are represented as coordinate vectors, then the linear functions are those functions that can be expressed as
- f(x) = Mx, where M is a matrix.
A function f(x) = mx + b is a linear map if and only if b = 0. For other values of b this falls in the more general class of affine maps.
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