STEP-BY-STEP LESSON · §7.2
One-to-One, Onto, and Inverse Functions
Distinguish injectivity, surjectivity, and existence of an inverse function.
Before you begin
Functions Defined on General Sets
X is the set of allowed inputs; Y is the declared set of possible outputs. Both are part of the function specification.
f:X→Y
Every x∈X must have exactly one image in Y. Different inputs may share an output.
∀x∈X ∃!y∈Y: y=f(x)
Symbols
- f⁻¹
- inverse function, when it exists
- ℝ
- the set of real numbers
- [0,∞)
- all nonnegative real numbers
Definitions and notation for this topic
Intervals and endpoints
[a,b]; (a,b); [0,∞)
For real a<b, [a,b] includes both endpoints; (a,b) excludes both. A square bracket includes its finite endpoint and a round bracket excludes it.
Infinity is not a real endpoint to include, so ∞ always has a round bracket in this notation. Intervals contain real values, not only integers.
Separate glossary example
[0,∞)={x∈ℝ:x≥0}; (−1,2]={x∈ℝ:−1<x≤2}.
Preimage of a set
f⁻¹(T)={x∈A:f(x)∈T}For any function f:A→B and T⊆B, f⁻¹(T) is the set of all inputs whose outputs lie in T. No bijectivity is required.
The result is a set and may be empty or have several elements. This use of f⁻¹ does not claim an inverse function exists.
Separate glossary example
For f:ℝ→ℝ, f(x)=x², f⁻¹({4})={−2,2} and f⁻¹({−1})=∅.
Injection / one-to-one
Different inputs always have different outputs.
Equivalently, f(a)=f(b) implies a=b.
For f:A→B, the full condition is ∀x,y∈A (f(x)=f(y)→x=y). The arrow ↣ is an optional convention in some texts, not required notation for this property.
Separate glossary example
f:ℤ→ℤ, f(x)=2x
Surjection / onto
Every element of the codomain is the output of at least one input.
For a target value y, the input y−1 works.
For f:A→B, onto means ∀b∈B ∃a∈A (f(a)=b). The codomain B matters. The arrow ↠ is an optional notation, not the definition.
Separate glossary example
f:ℤ→ℤ, f(x)=x+1
Bijection
A function f:A→B that is both injective and surjective. Every element of B has exactly one preimage in A.
Specify the function and its domain/codomain. A bare ↔ usually marks a logical biconditional here; it is not a sufficient notation for a bijection.
Separate glossary example
f:ℤ→ℤ, f(x)=x+1 is bijective: the unique input for y is x=y−1.
f:ℤ→ℤ, f(x)=x+1
Inverse function
f⁻¹:B→A
A bijection f:A→B has an inverse f⁻¹:B→A returning the unique input for each output. It satisfies f⁻¹(f(a))=a and f(f⁻¹(b))=b.
The domain and codomain restrictions make the inverse possible. f⁻¹(y) denotes inverse-function evaluation, not the reciprocal 1/f(y); a preimage f⁻¹(T) of a set remains meaningful without an inverse function.
Separate glossary example
f:[0,∞)→[0,∞), f(x)=x² has inverse f⁻¹(y)=√y. The same square rule on ℝ→ℝ is not bijective.
Step by step
Step 1 / 6
Check for collisions
Injectivity means equal outputs can come only from equal inputs.
f(x_1)=f(x_2) ⇒ x_1=x_2
This prevents an inverse from having to choose between two inputs.
Worked example
f:{1,2,3}→{a,b,c}, f(1)=b, f(2)=a, f(3)=c. Find the inverse.
- Outputs b,a,c are distinct: there are no collisions, so f is injective.
- Every element a,b,c is used, so f is surjective.
- Reverse the arrows: f⁻¹(a)=2, f⁻¹(b)=1, f⁻¹(c)=3.
Optional self-check
f:{1,2}→{a,b,c}, f(1)=a, f(2)=b: which properties hold?
Show answer
Injective but not surjective: c is not reached. There is no inverse function on all of {a,b,c}.