Calculus How To

Constant Function: Simple Definition & Example

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Types of Functions >

constant function

A constant function y(x) = 1.

A constant function is where the output variable (e.g. y) is not dependent on the input variable (e.g. x). Generally, it is a function which always has the same value no matter what the input is.

We can write this type of function as:

f(x) = c


  • c is a constant: a number that doesn’t change as x changes.

For example, the following are all constant functions:

  • f(x) = 99,
  • f(x) = .1 and
  • f(x)= -21

Since f(x) is equal to a constant, the value of f(x) will always be the same no matter what the value of x might be.

The Graph of a Constant Function

In a two dimensional plane, the graph of this type of function is a straight, horizontal line. It passes through the point (0, c), (1, c), and (-1, c). In fact, (a, c) for any a on the number line.

For example, the graph of the constant function f(x) = 4 is a straight, horizontal line that passes through the points (2,4), (0,4), and (-2,4). This graph is shown below.

constant function

Derivatives and Integrals

The derivative of this type of function is just zero. You can use direct substitution to arrive at this conclusion.

Every function with a derivative equal to zero is a constant function.

The integral of this type of function is equal to c x. For example, the integral of f(x) = 10 is 10x.

The following shows the graph of f(x) = 10, and the integral f(x) = 10x. The blue square represents the integral when evaluated from 0 to 1.
integral of a constant function

Limit of a Constant Function

The limit of a constant function (according to the Properties of Limits) is equal to the constant. For example, if the function is y = 5, then the limit is 5.

Applications of the Constant Function

Though it is one of the simplest type of functions, it can be used to model situations where a certain parameter is constant and isn’t dependent on the independent parameter. For instance, a school dining room where every child was given one donut, irrespective of age, or an exam in which every student was given an A regardless of how hard they worked.


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Stephanie Glen. "Constant Function: Simple Definition & Example" From Calculus for the rest of us!

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2 thoughts on “Constant Function: Simple Definition & Example

  1. Stephanie Post author

    Technically, zero is a constant. But 0/6 = 0, so this would not actually produce a graph. As a function requires that inputs produce outputs, it wouldn’t be a “function”.

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