Table of Content

• What is Constant Function?
• Characteristics of a Constant Function
• Graph of Constant Function
• Examples on Constant Function

Input (x)	Output (f(x))
R	c

Limit of Constant Function

The limit of a constant function as x approaches any value is simply the constant value itself. In other words, for a constant function i.e., f(x) = c, where c is a constant. 

lim _x→a f(x) = c

Where a can be any real number.

Derivatives of Constant Function

The derivative of a constant function is always equal to zero. In mathematical notation, if you have a constant function f(x) = c, where c is a constant value, then its derivative is:

f'(x) = 0

This means that the slope or the rate of change of a constant function is zero because it doesn’t change as “x” changes; it remains flat and constant.

Integral of Constant Function

The integral of a constant function is straightforward and can be expressed as follows:

If you have a constant function f(x) = c, where c is a constant value, then the integral of f(x) with respect to x is:

∫f(x) dx = ∫c dx = cx + C 

where,

• c is the constant value of the function, and
• C is the constant of integration.

So, when you integrate a constant function, you end up with a linear function with a slope equal to the constant value c, and the constant of integration C accounts for any vertical shift in the graph.

For example, if you want to find the integral of the constant function f(x) = 5, you would get:

∫ 5 dx = 5x + C

Graph of Constant Function

A constant function is characterised as a real-valued function that does not contain any variables, as has been previously mentioned. 

Graph of Constant Function can be represented by a horizontal line i.e., a line parallel to the x-axis.

To illustrate this concept, let us consider the example of the function f(x) = 5, where (f) maps from the set of real numbers to itself, i.e., f: R → 5. For this function, the output is always 5 for all real numbers, i.e., (-2, 5), (0, 5), (3, 5), and so on, all lie on the graph. The following illustration shows the graph of f(x) = 5.

Slope of Constant Function

As the slope of any graph is the value of the first derivative of that function at that point, and for the entire domain of the constant function, its derivative is 0. Thus,

The slope of a constant function is always equal to zero.

Note: For any line with the equation y = mx + b, ‘m’ represents the slope of the line. However, for a constant function, we can represent it as y = 0x + k. Thus, we can see that the constant function is a straight line with a slope of 0.

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Examples on Constant Function

Example 1: Find the derivative of the function f(x)=19.

Solution:

f(x) is a constant function with the value 19.

Thus, change in output with respect to x, is always be 0

Therefore: f'(x) = 0

Example 2: Find the limit of the constant function f(x) = 7 as x approaches any real number, say a.

Solution:

Limit of a constant function as x approaches any real number is simply the constant value itself.

lim _x→a 7 = 7

So, the limit of the constant function f(x) = 7 as x approaches any real number is 7.

Example 3: Given a constant function f(x) = -3, find the integral of this function over the interval [1, 5].

Solution:

∫₁⁵ (-3) dx = (-3) × [x]₁⁵

⇒ ∫₁⁵ (-3) dx = (-3) × (5 – 1)

⇒ ∫₁⁵ (-3) dx = (-3) × 4 = -12

So, the integral of the constant function f(x) = -3 over the interval [1, 5] is -12. [Where the negative sign shows the area being under the x-axis.]

Practice Problems on Constant Function

Problem 1: Determine whether the following function is a constant function:

• f(x) = 5
• g(x) = 3x
• h(x) = -1/2
• k(x) = 1-x

Problem 2: Find the constant value c such that the function g(x) = c passes through the point (3, 7).

Problem 3: Given a constant function h(x) = -2, calculate h(10).

Problem 4: Determine the constant k for which the function p(x) = k is a horizontal line with a slope of 0.

Problem 5: If the function q(x) = 9 is shifted 3 units upward, what is the new equation of the function?

Problem 6: Consider the function m(x) = 3 and the function n(x) = 2. Determine whether m(x) + n(x) is a constant function.

Problem 7: Answer the following question for the constant function f(x) = 8.

• What does f(3) represent?
• What is the function’s domain?
• What is the function f'(x)’s derivative?

Problem 8: For the constant function k(x) = -3, Explain the following questions.

• Find k(5).
• Identify k(x)’s domain and range.
• Calculate k(x), k'(x).

Conclusion

A constant function in mathematics is a function that always returns the same value, regardless of the input. Formally, a function [Tex]f: \mathbb{R} \to \mathbb{R}[/Tex] is called a constant function if there exists a constant c∈R such that for every x∈R, f(x)=c. This means that the graph of a constant function is a horizontal line in the Cartesian plane. Constant functions are important in calculus and algebra because they have a derivative of zero, indicating that they have no rate of change. They serve as basic examples in various mathematical contexts and are used in modeling situations where a quantity remains unchanged over time or across different conditions.

Constant Function FAQs

Define Constant Function.

A constant function, also known as a constant mapping or a constant transformation, is a mathematical function that assigns the same constant value to every input in its domain.

What is a Constant Function in Algebra?

No matter what values are inputted, a constant function will always yield the same outcome.

How Does Graph of Constant Function Look?

Graph of a constant function is a straight line parallel to the x-axis.

What is a Constant Function Equation?

The equation f(x) = c represents a constant function, where ‘c’ is a fixed numerical value such as 7. This means that regardless of the input ‘x’, the output is always same.

How do you Know if a Function is Constant?

A function is defined as constant when it lacks variables and possesses an unequivocal graph resembling a long, unswerving line that stretches horizontally.

Is a Constant Function Continuous?

As constant function represents the straight line parallel to the x-axis, a constant function is always continuous in its complete domain.

What is the Integral of a Constant Function?

The integral of a constant function can be expressed as kx + C, where ‘k’ represents the constant. For example, when integrating ∫3 dx, the result is 3x + C.

What is the Derivative of a Constant Function?

The derivative of any constant function is always zero.

Is Constant Function Injective?

No, it is not an injective since a constant function does not have a one-to-one relationship between input and output.

What is the Degree of Constant Function?

The degree of a constant function is zero because it can be written as f(x) = c * x⁰, where ‘c’ is the constant.

What is an Example of Constant Function?

An example of a constant function would be f(x)= 3.

Why is it called Constant Function?

It is called a constant function because of its nature and it provides a constant value in the output no matter what the input is provided.

Is Identity Function a Constant Function?

No, the identity function is not a constant function as it maps all the inputs to itself i.e., for any identity function f; f(1) = 1, f(2) = 2, and so on.