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Constant function facts for kids

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Constant function y=4

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

Basic properties

Formally, a constant function f(x):RR has the form  f(x)=c. Usually we write y(x)=c  or just  y=c.

  • 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  y(x)=4  or just  y=4  is the specific constant function where the output value is  c=4. 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 y=c is a horizontal line in the plane that passes through the point (0,c).
  • 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:  (c)'=0 .

Example:  y(x)=-\sqrt{2}  is a constant function. The derivative of y is the identically zero function   y'(x)=(-\sqrt{2})'=0  .

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:

y(x)=c  \,\,\, \Leftrightarrow \,\,\, y'(x)=0 \,, \,\,\forall x \in \mathbb{R}

Generalization

A function f : AB 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 : AB where A={X,Y,Z,W} and B={1,2,3} and f(a)=3 for every aA. 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.

Constant function gen2.svg
Generalized constant function.
Constant function plane.png
Constant function z(x,y)=2
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

Related pages

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