Constant Function in Mathematics

Constant Function is a specific type of mathematical function that, as its name suggests, outputs will always be the same value for any input. In other words, the output of a constant function remains constant. This is why it is called a constant function. Constant Function can be expressed simply as f(x) = a, where 'a' can be any real number that is completely independent of the dependent variable or input i.e., 'x'. The constant function is considered one of the most straightforward functions in mathematics.

This article provides a well-rounded description of the constant function as it covers all the important subtopics, such as the definition of a constant function, constant function notation, and characteristics like domain & range, limit, derivative, as well as integral. Additionally, this article on the constant function offers solved problems and answers to frequently asked questions to simplify complex mathematical concepts into easily understandable explanations.

What is Constant Function?

A constant function is a way to describe something that stays constant as time passes, and it's one of the most basic kinds of functions with real numbers. When you construct a graph with a constant function, you get a straight horizontal line. One practical example of a constant function is the fixed salary of an employee for every month.

Constant Function Definition

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.

Constant Function Notation

As output in the Constant Function Remain always the same, we can represent these function mathematically as:

y = f(x) = c

where,

• y or f(x) represents the output of the function, and
• c is the constant value that can be any real number.

Constant Function Example

There are various examples of Constant Functions. Some of them are listed below:

• Constant Zero Function: f(x) = 0
• Constant One Function: f(x) = 1
• Constant Negative Number Function: f(x) = -5
• Constant Positive Number Function: f(x) = 10
• Constant Pi Function: f(x) = π (pi)
• Constant e Function: f(x) = e
• Constant Square Root of 2 Function: f(x) = √2 (square root of 2)
• Constant Rational Number Function: f(x) = 3/4
• Constant Decimal Function: f(x) = 2.5
• Constant Fraction Function: f(x) = -1/3

Also Read: Calculus in Maths

Characteristics of a Constant Function

Constant functions always meet the vertical axis at a spot which is determined by their constant value. However, these functions never intersect with the horizontal axis since it runs parallel to them. Instead, they generate straight horizontal lines that extend infinitely in both directions, forming a continuous pattern. 

Now let's explore some significant aspects of constant functions:

• Domain and Range
• Limit
• Derivative
• Integral

Domain and Range of Constant Function

In the context of a constant function, it does not matter what real number you choose as "x"; the function will always give you the same result, the function doesn't change, and it will be the same no matter whatever the value of "x".

• Domain: The domain of a constant function is all real numbers i.e., Domain: (−∞,∞)
• Range: The range of a constant function is a singleton set containing only the constant value "c", i.e., Range: {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 f: \mathbb{R} \to \mathbb{R} 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.

Read More:

• List of Mathematical Constant
• What Is a Constant? Definition, Examples, Facts
• Constant Function in Mathematics