Types of Functions

The definitions

A function $f$ is one-one (injective) if different inputs always give different outputs, and onto (surjective) if every element of the codomain is hit. A function that is both is bijective, which is exactly the condition for it to be invertible.

Worked example: checking one-one and onto

Let $f: \mathbb{N} \to \mathbb{N}$ with $f(n) = \dfrac{n+1}{2}$ for odd $n$ and $\dfrac{n}{2}$ for even $n$. Since $f(1) = f(2) = 1$, $f$ is not one-one. For any $m$, $f(2m) = m$, so $f$ is onto.

Worked example: restricting the domain

$f(x) = x^2$ on $\mathbb{R}$ is not one-one, because $f(-3) = f(3) = 9$. Restricted to $[0, \infty)$, different inputs give different squares, so it is one-one, and as a map onto $[0, \infty)$ it is bijective, with inverse $\sqrt{x}$.

Worked example: finding an inverse

For $f: \mathbb{N} \to S$, $f(x) = 4x^2+12x+15$ with $S$ the range, complete the square: $f(x) = 4\left(x+\tfrac{3}{2}\right)^2 + 6$. Since $x + \tfrac{3}{2} > 0$ on $\mathbb{N}$, solving is reversible:

Check: $f(5) = 100 + 60 + 15 = 175$, and $\dfrac{\sqrt{169}-3}{2} = \dfrac{13-3}{2} = 5$.

Worked example: a reflection is an inverse

For $f(x) = (x+1)^2$ with $x \ge -1$, the reflection of its graph in $y = x$ is the inverse: $y = (x+1)^2 \Rightarrow x = \sqrt{y} - 1$, so $g(x) = \sqrt{x} - 1$.

Counting functions between finite sets

From a set of 3 elements to a set of 4, each input has 4 choices, so there are $4^3 = 64$ functions, of which $4 \times 3 \times 2 = 24$ are one-one. From a 4-element set onto a 3-element set, inclusion and exclusion gives $3^4 - 3 \cdot 2^4 + 3 \cdot 1^4 = 81 - 48 + 3 = 36$ onto functions.

Worked example: a linear function

Show that $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 3x - 5$ is bijective and find its inverse.

Solution: If $3a - 5 = 3b - 5$ then $a = b$, so $f$ is one-one. For any real $y$, $x = \dfrac{y+5}{3}$ is real and $f(x) = y$, so $f$ is onto. Hence $f^{-1}(y) = \dfrac{y+5}{3}$. The same argument works for any $f(x) = ax + b$ with $a \ne 0$.

Common mistakes

• Ignoring the codomain. Whether $f$ is onto depends on the stated codomain, not just the formula.
• Inverting a function that isn't one-one. Restrict the domain first.
• Taking both square roots when inverting. The domain decides which sign is correct.

Where this comes up in exams

The 2025-26 CBSE Class XII syllabus lists one to one and onto functions. JEE Main 2026 lists one-one, into and onto functions and the composition of functions, and JEE Advanced 2026 lists invertible functions and into, onto and one-to-one functions.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26), JEE (Main) 2026 syllabus (NTA) and JEE (Advanced) 2026 syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

• Fit it into your week with the free timetable generator.
• Mark it off as you revise in the syllabus tracker.
• See how long you have left: JEE Main 2027 (Session 1) countdown and JEE Advanced 2027 countdown.