MathematicsChapter 6: Functions and Graphs
Injective, Surjective, Bijective
One-to-one (injective): different inputs -> different outputs.
Learning objectives
- Explain injective, surjective, bijective using the correct definitions, conditions, and notation.
- Apply 1-1: outputs unique, onto: all targets hit correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
One-to-one (injective): different inputs -> different outputs. Onto (surjective): every element of the stated codomain is hit. Bijective: both, so an inverse function exists as a function the other way. Identity f(x)=x is bijective on R.
Where / symbols
- INJECTIVE, SURJECTIVE, BIJECTIVE
- One-to-one (injective): different inputs -> different outputs. Onto (surjective): every element of the stated codomain is hit. Bijective: both, so an inverse function exists as a function the other way. Identity f(x)=x is bijective on R.
- INJECTIVE
- If f(a)=f(b) then a=b. Horizontal line test on a graph: a horizontal line meets the graph at most once. x^2 on all reals is not injective because f(2)=f(-2).
- SURJECTIVE AND BIJECTIVE
- f: R -> R, f(x)=e^x is not onto (negative never hit). f(x)=2x+1 on R is onto and 1-1, hence bijective, hence invertible.
Overview
Two extra adjectives:
Injective
If f(a)=f(b) then a=b. Horizontal line test on a graph: a horizontal line meets the graph at most once. x^2 on all reals is not injective because f(2)=f(-2).
Examples x^2 on R not 1-1 · f(x)=2x+1 is 1-1 · Horizontal line test
Surjective and Bijective
f: R -> R, f(x)=e^x is not onto (negative never hit). f(x)=2x+1 on R is onto and 1-1, hence bijective, hence invertible.
Examples identity is bijective · Need both for inverse · Codomain matters
WHAT — INJECTIVE, SURJECTIVE, BIJECTIVE
One-to-one (injective): different inputs -> different outputs. Onto (surjective): every element of the stated codomain is hit. Bijective: both, so an inverse function exists as a function the other way. Identity f(x)=x is bijective on R. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Injective, Surjective, Bijective matters because functions describe how one quantity depends on another. Domain, range, composition, and inverse ideas are the language used for graphs and mathematical models.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through INJECTIVE, then SURJECTIVE AND BIJECTIVE. Worked: is f(x) = x^2, domain all reals, injective? f(2) = 4 and f(-2) = 4 two different inputs, same output so f is not injective on all reals Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- is f(x) = x^2, domain all reals, injective?
- f(2) = 4 and f(-2) = 4
- two different inputs, same output
- so f is not injective on all reals
In simple words
Changing the domain to x>=0 can make x^2 injective. The same formula with a smaller domain is a different function story.
Key points
- 1-1: outputs unique
- onto: all targets hit
- no shared output
- cover the codomain
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Injective, Surjective, Bijective
Sticky Notes · Quick Revision
Core Idea
- One-to-one (injective): different inputs -> different outputs. Onto (surjective): every element of the stated codomain is hit. Bijective: both, so an inverse function exists as a function the other way. Identity f(x)=x is bijective on R.
INJECTIVE
- If f(a)=f(b) then a=b. Horizontal line test on a graph: a horizontal line meets the graph at most once. x^2 on all reals is not injective because f(2)=f(-2).
SURJECTIVE AND BIJECTIVE
- f: R -> R, f(x)=e^x is not onto (negative never hit). f(x)=2x+1 on R is onto and 1-1, hence bijective, hence invertible.
Worked Method
- Worked: is f(x) = x^2, domain all reals, injective?
- f(2) = 4 and f(-2) = 4
- two different inputs, same output
- so f is not injective on all reals
Common Misconception
- Changing the domain to x>=0 can make x^2 injective. The same formula with a smaller domain is a different function story.
FBISE Exam Check
- Changing the domain to x>=0 can make x^2 injective. The same formula with a smaller domain is a different function story.
60-second revision
- One-to-one (injective): different inputs -> different outputs.
- INJECTIVE: If f(a)=f(b) then a=b.
- SURJECTIVE AND BIJECTIVE: f: R -> R, f(x)=e^x is not onto (negative never hit).
- Worked-method reminder: so f is not injective on all reals
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