Constant function

In mathematics, a constant function is a function whose output value is the same for every input value.   For example, the function  is a constant function because the value of    is 4 regardless of the input value  (see image).

Basic properties
Formally, a constant function f(x):R→R has the form  . Usually we write   or just  . 
The function y=c has 2 variables x and у and 1 constant c. (In this form of the function, we do not see x, but it is there.)
The constant c is a real number. Before working with a linear function, we replace c with an actual number.
The domain or input of y=c is R. So any real number x can  be input. However, the output is always the value c.  
The range of y=c is also R. However, because the output is always the value of c, the codomain is just c.

Example: The function    or just    is the specific constant function where the output value is  . The domain is all real numbers ℝ. The codomain is just {4}.  Namely, y(0)=4, y(−2.7)=4, y(π)=4,.... No matter what value of x is input, the output is "4".

The graph of the constant function  is a horizontal line in the plane that passes through the point .
If c≠0, the constant function y=c is a polynomial in one variable x of degree zero.
The y-intercept of this function is the point (0,c).
This function has no x-intercept. That is, it has no root or zero. It never crosses the x-axis.
If c=0, then we have y=0. This is the zero polynomial or the identically zero function. Every real number x is a root. The graph of y=0 is the x-axis in the plane. 
A constant function is an even function so the y-axis is an axis of symmetry for every constant function.

Derivative of a constant function
In the context where it is defined, the derivative of a function measures the rate of change of function (output) values with respect to change in input values. A constant function does not change, so its derivative is 0. This is often written:   .

Example:    is a constant function. The derivative of y is the identically zero function     .

The converse (opposite) is also true. That is, if the derivative of a function is zero everywhere, then the function is a constant function.

Mathematically we write these two statements: 

Generalization
A function f : A → B is a constant function if f(a) = f(b) for every a and b in A.

Examples
Real-world example: A store where every item is sold for 1 euro. The domain of this function is items in the store. The codomain is 1 euro.

Example: Let f : A → B where A={X,Y,Z,W} and B={1,2,3} and f(a)=3 for every a∈A. Then f is a constant function.

Example: z(x,y)=2 is the constant function from A=ℝ² to B=ℝ where every point (x,y)∈ℝ² is mapped to the value z=2. The graph of this constant function is the horizontal plane (parallel to the x0y plane) in 3-dimensional space that passes through the point (0,0,2).

Example: The polar function ρ(φ)=2.5 is the constant function that maps every angle φ to the radius ρ=2.5. The graph of this function is the circle of radius 2.5 in the plane.

180px Generalized constant function.180pxConstant function z(x,y)=2180pxConstant polar function ρ(φ)=2.5

Other properties
There are other properties of constant functions.

Related pages
Matrix
Calculus
Function
Arithmetic
Integration
Polynomial
Differentiation
Quadratic function
Differential equation
Multivariable calculus
List of mathematical symbols

References

Other websites

Category:Functions and mappings
Category:Algebra