# Constant function facts for kids

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).

## Contents

## 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*.

- The constant

**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.

- The
- 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 *x*0*y* 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.

Generalized constant function. |
Constant function z(x,y)=2 |
Constant polar function ρ(φ)=2.5 |

## Other properties

There are other properties of constant functions. See Constant function on English Wikipedia

## Related pages

## Images for kids

*Kiddle Encyclopedia.*