Linear Function

Calculus is based upon the linear Functions. Linear Functions are the functions whose graphs consist of sections of one straight line throughout the function's Domain.
Linear function Statistics can be defined as the functions that have ‘x’ as the input variable and ‘x’ has an exponent of only 1.
Linear functions can refer to the following concepts
·         A first degree polynomial function having one variable.
·         A map between two vector spaces that refers vector addition and Scalar multiplication.
Linear function are said to be linear because the graphs of these functions in the Cartesian co- ordinate plane is Straight Line.
For example there are some functions below whose graph is a straight line
·         g (x) = 2x + 4
·         g (x) = x/2 - 3
Linear functions can be written in the following format
f (x) = mx + b,
(y – y₁) = m (x - x₁),
0 = Ax + By + C,
In vector Algebra a linear function means a linear map i.e. a map between two vectors spaces that refers vector addition and Scalar Multiplication.
The linear functions are those functions ‘f’ that can be expressed as,
f (x) = Kx,
where ‘K’ is a matrix.
A function g (x) = mx + b is called a linear map if and only if b = 0.
The form y = mx + b is named as the 'slope intercept form' of linear function. For a common graph (x,y) usually expressed as y = mx + b and in a formal function definition a linear function written as f (x) = mx + b.
We can derive this fact for the linear functions. Suppose a linear function which takes value ‘g’ (c₁) at c₁ and g (c₂) at c₂ by the following formula-
g (x) = g (c₁) x – c₂ / c₁ – c₂ + g (b) x – c₁ / c₂ – c_1,
The first term will be zero when ‘x’ is ‘c_2’ and is ‘g (c₁)’ when ‘x’ is ‘c_1’, while the second term is zero when ‘x’ is ‘c_1’ and is g(c₂) when ‘x’ is ‘c_2’.
More convenient form for this function is-
g (x) = x g (c₂) – g (c₁) / c₂ – c₁] + [c₂ f(c₁) – c₁ f (c₂) / c₂ – c₁].
g (x) = mx + c,
Here the ‘m’ shows the Slope of this line. If we plot the graph on the ‘y’ axis then ‘c’ is called here the ‘y’ intercept of the line.
The description of this equation can be provided as a Set of or locus of (x, y) points and these points lie along a straight line. The variable m refers to the Slope of this line and the variable ‘b’ refers to the ‘y’ co-ordinate where the line crosses the y- axis named as 'y-intercept'.
Point slope form for the linear functions shows the equation of a line i.e.
(y – y₁) = m (x - x₁).