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One-one and onto questions look like definitions, so students expect easy marks. Then the answer depends on a codomain they did not read, or a proof skips a step the marking scheme wanted. This post gives you a fixed method for each test, worked examples that cover the usual cases, and the shortcuts that help in JEE.
Where do these ideas appear in the syllabus?
The CBSE Mathematics curriculum for 2026-27 lists "One to one and onto functions" in Unit I of Class XII, next to types of relations. The NCERT Class 12 chapter on relations and functions (reprint 2026-27) gives the definitions in section 1.3. For JEE, Unit 1 of the JEE (Main) 2026 syllabus includes "one-one, into and onto functions, the composition of functions". Check the syllabus for your own exam year.
What do one-one and onto mean?
For f : X → Y, NCERT defines:
One-one (injective): distinct elements of X have distinct images. Equivalently, f(a) = f(b) forces a = b.
Onto (surjective): every element of Y is the image of some element of X. NCERT's remark puts it neatly: f is onto exactly when the range of f equals Y.
Bijective: both one-one and onto.
Research on how students learn functions suggests why this feels slippery. Breidenbach and colleagues (1992) argued that many college students lack a working understanding of function and reported large improvements for many students when teaching treated a function as a process that takes an input and returns an output. That is the picture to hold here: one-one asks whether the process can ever merge two inputs; onto asks whether it reaches every target.
How do you prove a function is one-one or onto?
To prove one-one: start from f(a) = f(b) and reach a = b. To disprove it: give two different inputs with the same output.
To prove onto: take any y in the codomain, solve y = f(x) for x, and check that x lies in the domain. To disprove it: give one y with no preimage.
Worked example 1. f : R → R, f(x) = 3x − 5.
One-one: 3a − 5 = 3b − 5 gives a = b.
Onto: y = 3x − 5 gives x = (y + 5)/3, a real number for every real y.
So f is bijective.
Worked example 2. f : R → R, f(x) = x².
Not one-one: f(2) = f(−2) = 4.
Not onto: −1 has no preimage, since x² cannot be negative.
Change the sets to f : [0, ∞) → [0, ∞) and the same formula becomes bijective. The formula did not change; the domain and codomain did.
Worked example 3. f : R − {2} → R − {1}, f(x) = (x − 1)/(x − 2).
One-one: (a − 1)/(a − 2) = (b − 1)/(b − 2). Cross-multiplying, ab − 2a − b + 2 = ab − a − 2b + 2, so −2a − b = −a − 2b, which gives a = b.
Onto: y = (x − 1)/(x − 2) gives yx − 2y = x − 1, so x = (2y − 1)/(y − 1). This exists for every y ≠ 1. And x is never 2, because x = 2 would need 2y − 2 = 2y − 1, which is impossible.
So f is bijective.
What does the board marking scheme reward?
CBSE's Class XII Mathematics marking scheme for the 2026-27 sample paper has a five-mark question on f : (−∞, ∞) → [0, ∞), f(x) = x² + 5. It gives 1½ marks for showing the function is not injective, using f(−1) = f(1) = 6, and another 1½ marks for showing it is not surjective, because the range is [5, ∞) while the codomain is [0, ∞). One clear counterexample and one statement about the range earn those marks. Write them as separate, labelled parts.
Which shortcuts help in JEE?
Horizontal line test. On a graph, a function is one-one if no horizontal line cuts it twice. It is onto its codomain if every horizontal line at a codomain height cuts it at least once.
Monotonic functions. If f′(x) > 0 for all x, f is strictly increasing and therefore one-one. For f(x) = x³ + x, f′(x) = 3x² + 1 > 0, so f is one-one on R. It is also onto R, because it is continuous and takes arbitrarily large positive and negative values.
Counting. For a set A with m elements and B with n elements, there are nᵐ functions from A to B. If m ≤ n, the number of one-one functions is n!/(n − m)!. From {1, 2, 3} to {a, b} there are 2³ = 8 functions, and all except the 2 constant ones are onto, so 6 are onto.
Common mistakes and how to avoid them
Proving one-one with an example. One pair with different outputs proves nothing; you need the general argument.
Solving for x in the onto test and forgetting to check that x is in the domain.
Calling f(x) = x/(1 + x²) onto R. Its range is [−1/2, 1/2], and it is not one-one either: f(2) = f(1/2) = 2/5.
Mixing up range and codomain. If you need to revise how to find a range, our post on domain, range and composite functions walks through it.
A practice routine
Week 1: Ten functions, each tested for one-one and onto with the full written method. Include three where you change the codomain and the answer changes.
Week 2: Graph-based questions using the horizontal line test, plus five monotonic-function questions.
Week 3: Counting questions and mixed board-style questions, written to the marking scheme layout.
If the definitions themselves feel shaky, the Fermi Concept Library covers every concept in mathematics, each with its own lesson and its place in the map, so you can revise relations, functions and inverses in order. Our post on sets and types of relations covers the chapter that comes just before this one.
Take worked example 2 and write down two other domain and codomain pairs that make x² one-one but not onto.
