One-One, Onto, Bijective and Composite Functions | NIMCET

Quick answer: A function $f : A \to B$ is one-one (injective) if different inputs give different outputs, onto (surjective) if every element of B is an output, and bijective if it is both. Only bijective functions have an inverse. The composite $(g \circ f)(x) = g(f(x))$ applies f first, then g, and in general $g \circ f \ne f \circ g$ .

This continues from functions, domain, codomain and range.

Definitions and tests

Type	Definition	Quick test
One-one	$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$	Horizontal line meets the graph at most once
Many-one	Two inputs share an output	Some horizontal line meets the graph twice
Onto	Range = codomain	Every y in B has a solution x
Into	Range ⊂ codomain, range ≠ codomain	Some y has no pre-image
Bijective	One-one and onto	Both tests pass

A function that keeps increasing (or keeps decreasing) over its whole domain, with no flat stretch, is one-one; checking the sign of the derivative is a fast test.

Standard examples

Function	One-one	Onto
$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = 2x + 3$	Yes	Yes
$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x^2$	No	No
$f : [0, \infty) \to [0, \infty)$ , $f(x) = x^2$	Yes	Yes
$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x^3$	Yes	Yes
$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = e^x$	Yes	No
$f : \mathbb{R} \to \mathbb{R}$ , $f(x) = \sin x$	No	No

The same rule can change type when the domain or codomain changes, as $x^2$ shows.

Counting functions

For $n(A) = m$ and $n(B) = n$ :

Type	Number
All functions	$n^m$
One-one ( $m \le n$ )	$\frac{n!}{(n - m)!}$
Bijections ( $m = n$ )	$n!$
Onto ( $m \ge n$ )	$\sum_{r=0}^{n}(-1)^r\binom{n}{r}(n - r)^m$

Example: onto functions from a 4-element set to a 2-element set: $2^4 - 2 = 14$ (all functions minus the 2 constant ones).

Composite functions

$(g \circ f)(x) = g(f(x))$

This is defined when the range of f lies inside the domain of g.

Property	Statement
Not commutative	$g \circ f \ne f \circ g$ in general
Associative	$h \circ (g \circ f) = (h \circ g) \circ f$
One-one	If f and g are one-one, so is $g \circ f$
Onto	If f and g are onto, so is $g \circ f$

Example: $f(x) = x + 1$ and $g(x) = x^2$ . Then $(g \circ f)(x) = (x + 1)^2$ and $(f \circ g)(x) = x^2 + 1$ , which are different.

Inverse functions

If $f : A \to B$ is a bijection, its inverse $f^{-1} : B \to A$ satisfies $f^{-1}(y) = x \iff f(x) = y$ .

Method: write $y = f(x)$ , solve for x, then swap the letters.

Example: $f(x) = \frac{2x + 3}{5}$ .

Step 1: $5y = 2x + 3$ , so $x = \frac{5y - 3}{2}$ .

Step 2: $f^{-1}(x) = \frac{5x - 3}{2}$ .

Key facts:

$f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$ .
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ .
The graph of $f^{-1}$ is the reflection of the graph of f in the line $y = x$ .

Worked examples

Example 1: testing one-one

Is $f(x) = x^2 - 4x$ one-one on ℝ? No: $f(0) = f(4) = 0$ .

Example 2: testing onto

Is $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = 3x - 7$ onto? For any y, $x = \frac{y + 7}{3}$ is real, so yes.

Example 3: counting one-one functions

How many one-one functions are there from $\{1, 2, 3\}$ to $\{a, b, c, d\}$ ?

$\frac{4!}{1!} = 24$ .

Example 4: composite value

If $f(x) = 2x$ and $g(x) = x^2 + 1$ , find $(f \circ g)(2)$ .

$g(2) = 5$ , then $f(5) = 10$ .

Summary table

Concept	Key result
One-one	Distinct inputs give distinct outputs
Onto	Range equals codomain
Bijection	Has an inverse
One-one count	$\frac{n!}{(n - m)!}$
$(g \circ f)^{-1}$	$f^{-1} \circ g^{-1}$

Exam tips

Always check the stated domain and codomain before deciding onto or one-one.
For onto questions, try to solve $f(x) = y$ for a general y.
For two-element codomains, onto count $= 2^m - 2$ .
Read the order of composition carefully: $g \circ f$ means f first.

Continue with mathematical logic and truth tables, or see the NIMCET Set Theory chapter.

Common mistakes

Deciding onto without looking at the codomain.
Writing $(g \circ f)^{-1} = g^{-1} \circ f^{-1}$ ; the order reverses.
Assuming every one-one function is onto.
Confusing $f^{-1}(x)$ with $\frac{1}{f(x)}$ .

Frequently Asked Questions

What is a one-one function?

A function is one-one if different inputs always give different outputs, that is, f(x₁) = f(x₂) implies x₁ = x₂.

What is an onto function?

A function f : A → B is onto if every element of B is the image of at least one element of A.

How many bijections are there between two sets of n elements?

There are n! bijections.

When does a function have an inverse?

A function has an inverse only when it is bijective (one-one and onto).

Is fog equal to gof?

Not in general. Composition of functions is not commutative.