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
Where:
- 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.
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.
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.
References
Maplesoft Support. Online Help: Math Apps: Functions and Relations.Retrieved from https://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps%2FConstantFunction on May 24, 2019.
Stephanie Glen. "Constant Function: Simple Definition & Example" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/constant-function/
Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!
0/6 is a constant function
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”.