One-One, Onto and Bijective Functions
A function f: A to B is one-one if different elements of A have different images in B. It is onto if every element of B has a preimage in A. It is bijective if it is both one-one and onto.
In one line: One-one protects inputs from sharing an output; onto covers the full codomain.
Practice This ConceptUnderstand it
One-one depends on uniqueness of input for each output, while onto depends on whether the whole codomain is covered. The codomain must be stated before testing onto. A graph may help for real-valued functions: a horizontal line cutting the graph more than once indicates the function is not one-one.
Example
Formula and rule
One-one test: f(a) = f(b) implies a = b.
Onto test: for every y in codomain B, there exists x in domain A such that f(x) = y.
Bijective: one-one and onto.
Horizontal-line test: any horizontal line meeting the graph more than once means not one-one.
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
Check f: R to R, f(x) = x3 + 1.
- 1One-one: assume f(a) = f(b).
- 2Then a3 + 1 = b3 + 1, so a3 = b3, hence a = b.
- 3Therefore f is one-one.
- 4Onto: take any y in R.
- 5Put y = x3 + 1.
- 6Then x3 = y - 1, so x = cube root of (y - 1), which is real.
- 7Hence every real y has a real preimage, so f is onto.
- 8Therefore f is bijective.
Method you can reuse
- 1Write domain and codomain first.
- 2To prove one-one, put f(a) = f(b) and simplify until a = b. To disprove, give two different inputs with the same image.
- 3To prove onto, put y = f(x), solve for x, and verify x belongs to the domain.
- 4Combine the two results to decide bijective.
Sketch it in your answer
Use a mapping diagram or graph with axes. For graph-based checking, label x-axis, y-axis, graph y = f(x), and a horizontal line y = c. The horizontal-line test supports one-one decisions.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
Is f: R to R given by f(x) = x2 one-one?
Think first, then show answerHide answer
Common wrong answer
Claiming f(x) = x2 is onto from R to R because x = square root of y, while ignoring that negative y values in the codomain have no real preimage.
Don't mix this up
Exam tip
Write it in the exam
Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.
How CBSE asks it
- 1
Enough for 1 mark
A function f: A to B is one-one if different elements of A have different images in B. It is onto if every element of B has a preimage in A. It is bijective if it is both one-one and onto.
- 2
Add for 2 marks
Example: The function f: R to R defined by f(x) = 2x + 3 is one-one because f(a) = f(b) gives 2a + 3 = 2b + 3, so a = b. It is onto because for any y in R, y = 2x + 3 gives x = (y - 3)/2, which is real.
- 3
Add for 3 marks
One-one depends on uniqueness of input for each output, while onto depends on whether the whole codomain is covered.
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