Types of Functions: Identity, Constant, Polynomial & Rational

Once you know what a function is , the next step is meeting the specific “named” functions that appear again and again in higher mathematics. Four of the most fundamental types — identity, constant, polynomial, and rational functions — are introduced in this chapter, each with its own shape, domain, and range. Recognising these on sight saves time in exams and builds intuition for every function-related chapter that follows.

The Identity Function

The identity function is the simplest possible function: it returns exactly the value put in. Defined as f: R → R by f(x) = x for every x ∈ R, it leaves every input unchanged.

Both domain and range are R, the set of all real numbers. Its graph is a straight line through the origin, rising at 45 degrees, since output always equals input. At x = 3, the corresponding y-value is also 3, and the same holds for every real number.

The Constant Function

A constant function ignores its input entirely and always produces the same fixed output: f: R → R by f(x) = c, where c is some fixed real number, for every x ∈ R.

The domain is still all of R, but the range shrinks to a single value: {c}. Graphically, this is a horizontal line parallel to the x-axis, at height c. For example, f(x) = 3 produces a flat horizontal line through y = 3, regardless of x.

Polynomial Functions

A function f: R → R is a polynomial function if, for every x in R, it can be written as y = f(x) = a₀ + a₁x + a₂x² + … + aₙxⁿ, where n is a non-negative integer and a₀, a₁, …, aₙ are real numbers.

In simple terms, a polynomial function is any expression built from non-negative integer powers of x, combined with real coefficients. Functions like f(x) = x³ - x² + 2 and g(x) = x⁴ + √2·x qualify. But h(x) = x^(2/3) + 2x is not a polynomial function, since it involves a fractional power of x.

Special Cases Within Polynomial Functions

The identity function and constant function are both special cases of polynomial functions — identity is the polynomial f(x) = x (degree 1), and constant is a polynomial of degree 0. A linear function, f(x) = mx + c, is another commonly used polynomial, where m and c are fixed constants and the graph is always a straight line.

Graphing a Polynomial Function

Consider f(x) = x². Plotting values — f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 4 — reveals a curve dipping to a minimum at the origin and rising symmetrically, forming a parabola. The domain is all of R, while the range is restricted to {y : y ≥ 0}, since squaring never produces a negative result.

A cubic polynomial like f(x) = x³ behaves differently — negative for negative x, positive for positive x — and both its domain and range cover all of R.

Rational Functions

A rational function is formed by dividing one polynomial function by another, written as f(x)/g(x), where both f(x) and g(x) are polynomial functions of x, and the function is only defined wherever g(x) ≠ 0.

The most well-known example is f(x) = 1/x, defined for every real number except 0. This restriction exists because division by zero is undefined in mathematics. The domain of this function is R - {0}, and interestingly, the range is also R - {0}, since 1/x can equal any real number except zero, no matter how large or small x becomes.

The graph of f(x) = 1/x is a smooth curve called a hyperbola, split into two separate branches — one in the region where x is positive and y is positive, and another where x is negative and y is negative. As x approaches 0, the curve shoots off toward infinity, and as x grows very large in either direction, the curve flattens and approaches the x-axis without ever touching it.

Why These Function Types Matter

These four function families are not just abstract definitions — they form the toolkit used throughout Class 11 and 12 to model everything from straight-line motion (linear/identity functions), fixed costs in business mathematics (constant functions), area and volume relationships (polynomial functions), and rates or ratios that break down at specific values (rational functions). Recognising the shape of a function immediately from its equation is a skill that pays off well beyond this chapter.

Quick Recap

• Identity function: f(x) = x; domain = range = R; graph is a straight line through the origin.

• Constant function: f(x) = c; domain = R; range = {c}; graph is a horizontal line.

• Polynomial function: sum of non-negative integer powers of x with real coefficients.

• Rational function: f(x)/g(x), defined only where g(x) ≠ 0.

Related Reading

• What Is a Function in Maths? Definition, Examples & Function vs Relation

• Modulus, Signum & Greatest Integer Functions Explained

• Algebra of Real Functions: Domain, Range & Solved Examples

Practice What You’ve Learned

Identifying a function’s type at a glance, and sketching its graph correctly, is a skill examiners test repeatedly across CBSE boards and entrance exams like CUET and IPMAT. ChampionsPrep offers structured question banks, worked solutions, and mock tests for Class 11–12 Commerce students and competitive exam aspirants. Registration is free, and you only pay as you use the platform.

Test Your Knowledge

Q1.The Identity Function f: R → R defined by f(x) = x has:

Domain = R and Range = {0}Domain = R and Range = RDomain = (0, ∞) and Range = RDomain = Integers and Range = Rationals

Q2.For a constant function f: R → R defined by f(x) = c for all x ∈ R, the range consists of:

The entire real line RThe open interval (−c, c)A singleton set {c}The set of positive integers

Q3.The domain of the rational function f(x) = 1 / (x² − 9) is:

R − {9}(−3, 3)[3, ∞)R − {−3, 3}