What Are Constant and Identity Functions?

Functions play a foundational role in many mathematical topics, providing a formal mechanism to describe relationships between elements of two sets. Among the various types, constant and identity functions are fundamental cases frequently examined in calculus and algebra as well as in the context of Functions And Its Types. Their properties, graphical representations, and algebraic structures are essential for a comprehensive understanding of mathematical functions at the JEE level.

Definition of Constant Function

A constant function is a real-valued function in which every input is assigned the same output, regardless of the input variable. Formally, for a function $f: \mathbb{R} \to \mathbb{R}$, if there exists $c \in \mathbb{R}$ such that $f(x) = c$ for all $x \in \mathbb{R}$, then $f$ is a constant function.

Mathematically, the functional expression is given as $f(x) = c$ for every $x$ in the domain, where $c$ is a fixed real constant. This structural simplicity makes the constant function a specific instance of a polynomial function of degree zero.

The domain of a constant function is the set of all real numbers unless specified otherwise, and its range is the singleton set $\{c\}$.

Graphical Representation of Constant Function

The graph of $f(x) = c$ is a horizontal straight line parallel to the $x$-axis at the height $y = c$. Each point on this line has an ordinate equal to $c$, illustrating that the output remains unchanged for every input value.

This graphical property implies that the slope of the constant function is always zero, as there is no variation in the $y$-coordinate with respect to changes in $x$.

Properties of Constant Function

A constant function has several distinguished properties relevant for mathematical problem-solving and conceptual understanding. The following statements summarize fundamental attributes:

• Domain is $\mathbb{R}$ unless specified
• Range is the singleton set $\{c\}$
• Graph is a horizontal line
• Slope is zero everywhere
• Function is continuous over $\mathbb{R}$
• Every constant function is both many-to-one and surjective onto $\{c\}$

The constant function is also a continuous function, since for any $\varepsilon > 0$, it is possible to find a $\delta > 0$ such that $|f(x) - f(x_0)| < \varepsilon$, which is trivial because $f(x) = f(x_0)$ for all $x$.

Examples of Constant Function

Let $f(x) = 3$ for all $x \in \mathbb{R}$. This defines a constant function where the output is always $3$, irrespective of the input.

Similarly, $g(x) = \pi$ or $h(x) = -7$ are constant functions for respective values of $x$. The range of these functions is the respective singleton set, namely $\{3\}$ for $f(x)$ and $\{\pi\}$ for $g(x)$.

Definition of Identity Function

An identity function is a function that maps each element of its domain to itself. Explicitly, an identity function on $\mathbb{R}$ is defined as $I: \mathbb{R} \to \mathbb{R}$, $I(x) = x$ for all $x \in \mathbb{R}$.

This mapping ensures that every real number is mapped to itself, so the set of outputs (the range) is exactly equal to the set of inputs (the domain). The identity function is denoted by the symbol $I$ or sometimes by $f(x) = x$.

Structure and Notation of the Identity Function

Given any nonempty set $A$, the identity function $I_A$ on $A$ is defined by $I_A(x) = x$ for all $x \in A$. Therefore, the function preserves every element of its input set, acting as a neutral element in the composition of functions.

For example, for $A = \{1, 2, 3, 4, 5\}$, the identity function $I_A$ is the set of ordered pairs $\{(1,1),\ (2,2),\ (3,3),\ (4,4),\ (5,5)\}$.

Graph of the Identity Function

The graph of the identity function $f(x) = x$ on the cartesian plane is a straight line passing through the origin $(0, 0)$. It makes an angle of $45^\circ$ with both the $x$-axis and $y$-axis, and every point has coordinates of the form $(x, x)$.

This unique positioning distinguishes the identity function from all other linear functions with non-unit slope, as it bisects the first and third quadrants symmetrically.

Properties of Identity Function

The identity function possesses several distinctive mathematical properties:

• Domain and range are identical sets
• It is a bijective (one-to-one and onto) function
• The inverse of the identity function is itself
• Graph is the line $y = x$
• It is a linear function with slope $1$
• Continuous and differentiable everywhere on $\mathbb{R}$
• Acts as the identity element under function composition

Formally, for any function $f: A \to B$, the identity function satisfies $f \circ I_A = f$ and $I_B \circ f = f$ for all inputs. This property is central in the algebraic theory of functions and for Algebra Of Functions.

Difference Between Constant and Identity Function

Constant and identity functions exhibit fundamentally different mappings from the domain to the codomain. The following table highlights the critical distinctions:

Special Cases and Related Concepts

If a constant function is defined from a set $A$ to a codomain $\{c\}$, it is surjective. However, it is never injective except when the domain is a singleton. In contrast, the identity function is always injective and surjective if defined as $I_A: A \to A$.

Both constant and identity functions are continuous everywhere on $\mathbb{R}$, making them examples of the simplest classes of continuous functions. Their algebraic structures are key foundational references for general function classification and analysis as covered in Sets Relations And Functions.

Solved Examples

Consider $f(x) = 4$ for $x \in \mathbb{R}$. The domain is $\mathbb{R}$ and the range is $\{4\}$. For inputs $x = 0,\ -3,\ 2\sqrt{2}$, the output is always $4$.

Analyze the function $g(x) = x$ for $x \in \mathbb{R}$. For inputs $x = -5,\ 0,\ \pi$, the output matches the input, that is, $g(-5) = -5$, $g(0)=0$, $g(\pi)=\pi$. The range is $\mathbb{R}$.

Test injectivity for $h(x) = c$: If $h(x_1) = h(x_2)$ for any $x_1, x_2$, all inputs yield $c$, so it is not one-to-one (unless the domain has only one element). For $f(x) = x$, $f(x_1)=f(x_2)$ implies $x_1=x_2$, so it is injective.

Applications and Further Insights

Constant functions are frequently encountered in mathematical modeling where a parameter remains unaffected by the input, such as equilibrium concentrations or rates in physical and biological systems. Their stability makes them useful as reference or comparison baselines.

The identity function serves as a structural foundation in function composition and abstract algebra. In the algebra of functions, it is the neutral element under composition, satisfying $f \circ I = I \circ f = f$ for any composable function $f$. For more advanced usage, see Properties Of Determinants.

A comprehensive mastery of constant and identity functions lays the groundwork for deeper studies of injective, surjective, bijective, linear, and polynomial functions, and is essential for both conceptual clarity and problem solving in calculus and abstract algebra at the JEE level.