Types of Functions

Test your ability to identify and distinguish one‑one, many‑one, onto, and bijective functions from definitions, diagrams, and examples

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Views #: 299

Questions #: 20

Time: 20 minutes

Pass Score: 80.0%

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One-One (Injective) Function — Definition

1 pts

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The diagram below shows a mapping from set A to set B.

Which property does this mapping illustrate?

Many-one: two domain elements share an image

One-one (injective): distinct domain elements map to distinct images

Onto (surjective): every codomain element is an image

Bijective: it is both one-one and onto

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Explanation

Symbolic Condition for a One-One Function

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A function \(f: A \to B\) is one-one (injective) if and only if:

\(f(a_1) \ne f(a_2) \Rightarrow a_1 = a_2\)

\(f(a_1) = f(a_2) \Rightarrow a_1 = a_2\) for every \(a_1, a_2 \in A\)

\(a_1 = a_2 \Rightarrow f(a_1) \ne f(a_2)\)

Every element of B has exactly two pre-images in A

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Explanation

Many-One Function — Identify From a Diagram

1 pts

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Examine the mapping below.

What type of function is shown?

One-one (injective)

Many-one

Onto (surjective)

Bijective

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The Constant Function Is a Special Case Of …

1 pts

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The diagram below shows a constant function.

(diagram would go here)

A constant function is a special case of which type?

One-one (injective) function

Onto (surjective) function

Many-one function

Bijective function

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Explanation

Onto (Surjective) Function — Identify From a Diagram

1 pts

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Study the diagram below.

Which property does this mapping demonstrate?

One-one: all images are distinct

Onto (surjective): every codomain element has at least one pre-image

Bijective: it is one-one and onto

Not a function: one domain element has two images

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Explanation

Range and Codomain in an Onto Function

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For an onto (surjective) function \(f: A \to B\), which of the following is always true?

The range of f is a proper subset of B

The range of f equals the codomain B

The codomain B is a proper subset of the range

The range has fewer elements than the domain

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Explanation

Bijective Function — Identify From a Diagram

1 pts

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Examine the mapping shown.

What kind of function is this?

One-one but not onto

Onto but not one-one

Bijective (one-one and onto)

Many-one

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Why Are Bijective Functions Called 'Most Important'?

1 pts

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Bijective functions are described as the most important type. Which statement best explains why?

They are the easiest to compute and visualise

Each image corresponds to exactly one domain element, and every codomain element is involved

They always have a simple algebraic formula

They require the domain and codomain to be the same set

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Explanation

Conditions for a Relation to Be a Function

2 pts

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A relation \(f: A \to B\) is called a function when two conditions hold. Which of the following are those two conditions? (Select both.)

Each element of A is involved in the relation (no element of A is left without an image)

Each element of A is associated to exactly one element of B (not more than one)

Each element of B is involved in the relation

No element of B has more than one pre-image

The domain and codomain must be equal sets

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Is f(x) = x² Injective on ℝ?

1 pts

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Consider \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\). Is this function one-one (injective)?

Yes — every output value is the square of exactly one positive number

No — both x = 2 and x = −2 give f(x) = 4, so two distinct inputs share an output

Yes — because x² is always non-negative

No — because it is not defined for negative x

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Is f(x) = x² Surjective on ℝ?

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Consider \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^2\). Is this function onto (surjective)?

Yes — every real number can be expressed as x² for some x ∈ ℝ

No — negative numbers in ℝ cannot be outputs of x²

Yes — because f is defined for every x ∈ ℝ

No — because f is many-one

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Is f(x) = x³ Bijective on ℝ?

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Consider \(f: \mathbb{R} \to \mathbb{R}\) defined by \(f(x) = x^3\). What type is it?

Many-one and not surjective

One-one but not onto

Bijective (one-one and onto)

Onto but not one-one

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Is f(x) = eˣ Injective?

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Consider \(f: (-\infty, \infty) \to (0, \infty)\) defined by \(f(x) = e^x\). Is this function one-one (injective)?

No — many values of x give the same output

Yes — if e^x₁ = e^x₂ then x₁ = x₂, so distinct inputs give distinct outputs

No — e^x can equal 0 for some x

Yes — only because the codomain is restricted to positive numbers

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Identify the Function Type From a Diagram

1 pts

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Look at the mapping below — four domain elements map to three codomain elements.

What type of function is shown?

One-one (injective) — distinct inputs give distinct outputs

Bijective — one-one and onto

Many-one and onto — not injective, but surjective

Not a function — an element has two images

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Diagram — Is This Function Onto?

1 pts

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Examine the mapping below carefully.

Is this function onto (surjective)?

Yes — every domain element has an image

No — element 3 in the codomain has no pre-image in A

Yes — the function maps 3 elements to 3 elements

No — element a has two images

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Explanation

Match Each Function Type to Its Definition

2 pts

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To complete the line match

Click on an item in the first group
Click on an item in the second group
To delete a match, double click on a line

Type

Injective (one-one)

Surjective (onto)

Bijective

Many-one

Property

Every codomain element is the image of at least one domain element

Distinct domain elements always map to distinct codomain elements

Both injective and surjective; each image has exactly one pre-image

At least one codomain element is the image of two or more domain elements

Match each function type to its defining property.

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Classify These Functions

3 pts

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Function

×f(x) = x³

×f(x) = x²

×f(x) = 2x + 1

×f(x) = sin x

Injective only

Surjective only

Bijective

Neither

Classify each function as Injective only, Surjective only, Bijective, or Neither when viewed as functions from \(\mathbb{R}\) to \(\mathbb{R}\).

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Two Restrictions That Define Special Function Types

1 pts

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The general definition of a function places two conditions only on elements of the domain. If we impose analogous restrictions on elements of the codomain, which two special function types result?

Constant function and polynomial function

One-one (injective) function and onto (surjective) function

Finite function and infinite function

Continuous function and discontinuous function

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Sort Function Types by Restrictiveness

2 pts

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Please drag and drop the options to sort them

Sort these function types from least restrictive (top) to most restrictive (bottom).

Bijective (one-one and onto)

General function (no extra restriction on codomain)

Injective (one-one only)

Surjective (onto only)

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Is a One-One Function Always Onto?

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True or False: A one-one (injective) function \(f: A \to B\) is always also onto (surjective).

True

False

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Year 10

One-One (Injective) Function — Definition

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Explanation

Symbolic Condition for a One-One Function

Correct Answer

Explanation

Many-One Function — Identify From a Diagram

Correct Answer

Explanation

The Constant Function Is a Special Case Of …

Correct Answer

Explanation

Onto (Surjective) Function — Identify From a Diagram

Correct Answer

Explanation

Range and Codomain in an Onto Function

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Explanation

Bijective Function — Identify From a Diagram

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Explanation

Why Are Bijective Functions Called 'Most Important'?

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Explanation

Conditions for a Relation to Be a Function

Correct Answer

Explanation

Is f(x) = x² Injective on ℝ?

Correct Answer

Explanation

Is f(x) = x² Surjective on ℝ?

Correct Answer

Explanation

Is f(x) = x³ Bijective on ℝ?

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Explanation

Is f(x) = eˣ Injective?

Correct Answer

Explanation

Identify the Function Type From a Diagram

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Explanation

Diagram — Is This Function Onto?

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Explanation

Match Each Function Type to Its Definition

Type

Property

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Explanation

Classify These Functions

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Explanation

Two Restrictions That Define Special Function Types

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Explanation

Sort Function Types by Restrictiveness

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Explanation

Is a One-One Function Always Onto?

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Explanation

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