Constant function
In mathematics, a constant function is a function whose output value is the same for every input value.[1][2][3] For example, the function [math]\displaystyle{ y(x) = 4 }[/math] is a constant function because the value of [math]\displaystyle{ y(x) }[/math] is 4 regardless of the input value [math]\displaystyle{ x }[/math] (see image).
Basic properties
Formally, a constant function f(x):R→R has the form [math]\displaystyle{ f(x)=c }[/math]. Usually we write [math]\displaystyle{ y(x)=c }[/math] or just [math]\displaystyle{ y=c }[/math].
- 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 [math]\displaystyle{ y(x)=4 }[/math] or just [math]\displaystyle{ y=4 }[/math] is the specific constant function where the output value is [math]\displaystyle{ c=4 }[/math]. 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 [math]\displaystyle{ y=c }[/math] is a horizontal line in the plane that passes through the point [math]\displaystyle{ (0,c) }[/math].[4]
- 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.[5]
- 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.[6] This is often written: [math]\displaystyle{ (c)'=0 }[/math] .
Example: [math]\displaystyle{ y(x)=-\sqrt{2} }[/math] is a constant function. The derivative of y is the identically zero function [math]\displaystyle{ y'(x)=(-\sqrt{2})'=0 }[/math] .
The converse (opposite) is also true. That is, if the derivative of a function is zero everywhere, then the function is a constant function.[7]
Mathematically we write these two statements:
- [math]\displaystyle{ y(x)=c \,\,\, \Leftrightarrow \,\,\, y'(x)=0 \,, \,\,\forall x \in \mathbb{R} }[/math]
Generalization
A function f : A → B is a constant function if f(a) = f(b) for every a and b in A.[8]
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.
| File:Constant function gen2.svg Generalized constant function. |
File:Constant function plane.png Constant function z(x,y)=2 |
File:Constant function polar.png Constant polar function ρ(φ)=2.5 |
Other properties
There are other properties of constant functions. See Constant function on English Wikipedia