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Quizzes > Mathematics > Algebra

Questions for Functions Quiz: Test Your Relations Skills!

Think you can identify which of the relations are functions? Dive in now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration of interconnected nodes arrows and graphs on sky blue background for functions and relations quiz

Are you ready to take on questions for functions and elevate your understanding of relations? In this quiz, you'll dive into practice relations and functions exercises, discovering which of the relations are functions and determining is the relationship a function through real examples. Whether you're revisiting fundamentals with our foundations of functions quiz or looking to build on your algebra roots with a quick algebra quiz , every question guides you toward mastery. Perfect for students and self-learners, our challenge motivates critical thinking. Start now to test your knowledge and see how far you can go!

What is the formal definition of a function from set A to set B?
A relation where every element in A is related to at least one element in B.
A relation where each element in B is related to exactly one element in A.
A subset of the Cartesian product B×A.
A relation where every element in A is related to exactly one element in B.
A function f from A to B is defined as a subset of A×B such that every element of A appears exactly once as the first component of an ordered pair in f. This ensures each input has exactly one output. It is the foundational concept in mapping elements of one set to another. Further reading.
Given f = {(1,2),(2,3),(3,4)}, what is the domain of f?
{2}
{1, 2, 3, 4}
{1, 2, 3}
{2, 3, 4}
The domain of a function consists of all first components (inputs) of its ordered pairs. Here, the pairs are (1,2), (2,3), and (3,4), so the inputs are 1, 2, and 3. Thus the domain is {1, 2, 3}. Further reading.
If f(x) = 2x + 1 mapping from real numbers to real numbers, what is the range of f?
All real numbers less than 1.
All real numbers.
All real numbers greater than 1.
All real numbers greater than or equal to 1.
A linear function of the form f(x)=2x+1 has no restrictions on output as x varies over all real numbers. Its slope is nonzero, so it covers every real value exactly once. Therefore its range is all real numbers. Further reading.
Which of the following sets of ordered pairs defines a function?
{(1,2), (2,3), (3,2)}
{(3,4), (4,5), (3,6)}
{(1,2), (1,3), (2,3)}
{(1,1), (2,1), (2,2)}
A set of ordered pairs is a function if no input appears more than once with different outputs. In option B each first component 1, 2, and 3 appears exactly once. Options A, C, and D have repeated inputs mapping to different outputs, so they are not functions. Further reading.
What is the domain of f(x) = ?(x ? 3)?
x > 3
x ? 3
x ? 0
x ? 3
The expression under a square root must be nonnegative. Setting x?3 ? 0 gives x ? 3. Therefore the domain is all real x such that x is at least 3. Further reading.
Which term describes the set of possible outputs of a function?
Domain
Codomain
Image
Range
The range of a function is the set of all actual outputs the function can produce. The codomain may include values not actually attained, while the image and range are often used interchangeably, but range refers specifically to attained outputs. Further reading.
For the function f: {a, b, c} ? {1, 2, 3} given by f(a)=2, f(b)=3, f(c)=3, what is the image of b?
Undefined
1
2
3
The image of an element under a function is its output value. Here f maps b to 3, so the image of b is 3. Further reading.
Every one-to-one function has an inverse that is also a function.
False
True
A one-to-one (injective) function can be reversed uniquely for each output, so its inverse relation satisfies the definition of a function. This holds when the original function's codomain is restricted to its range. Further reading.
A function f: A ? B is one-to-one if which condition holds?
The range equals the codomain.
If f(a1) = f(a2) then a1 = a2.
For every b in B, there exists a in A with f(a) = b.
Every element of A is mapped to exactly one element of B.
Injectivity (one-to-one) means no two distinct inputs map to the same output. Formally, f(a1)=f(a2) implies a1=a2. This ensures uniqueness of mapping in the forward direction. Further reading.
A function f: A ? B is onto if which condition holds?
For every a in A, there is b in B with f(a) = b.
The domain equals the codomain.
For every b in B, there is a in A with f(a) = b.
No two inputs map to the same output.
Surjectivity (onto) means every element in the codomain B is an image of at least one element in A. Thus for each b in B, there exists some a in A with f(a)=b. Further reading.
Which condition guarantees that a function f: A ? B has an inverse f?¹ which is also a function?
f is one-to-one.
f is onto.
f is continuous.
f is bijective.
A function has a two-sided inverse if and only if it is both injective (one-to-one) and surjective (onto), which is the definition of bijectivity. Only bijections have inverses that are functions. Further reading.
Given f(x) = x + 1 and g(x) = 2x, what is (f ? g)(3)?
7
5
8
9
Composition f?g means apply g first, then f. Compute g(3)=6, then f(6)=6+1=7. Thus (f?g)(3)=7. Further reading.
If f(x) = x² and g(x) = x + 2, what is (g ? f)(x)?
x² + 4
(x + 2)²
2x² + 2
x² + 2
Compute f first: f(x)=x². Then apply g: g(f(x)) = f(x) + 2 = x² + 2. Thus (g?f)(x)=x²+2. Further reading.
Which of the following defines an equivalence relation on the integers?
a ~ b if a ? b is even.
a ~ b if a < b.
a ~ b if a + b = 0.
a ~ b if a · b > 0.
An equivalence relation must be reflexive, symmetric, and transitive. 'a?b is even' satisfies all three properties on ?. Other options fail at least one property. Further reading.
If f: X ? Y and g: Y ? Z, what is the domain of the composition g ? f?
X
Z
f(Y)
Y
The composition g?f maps elements of X first through f and then g, so its inputs come from X. Thus the domain of g?f is X. Further reading.
What is the range of the function f(x) = x² on the domain [?2, 3]?
[0, 9]
[2, 6]
[0, 4]
[1, 9]
On [?2,3], x² achieves its minimum 0 at x=0 and its maximum 9 at x=3. Thus the range is the interval [0,9]. Further reading.
Let f: A ? B be a bijection. What can be said about its inverse f?¹: B ? A?
It exists and is a function.
It is injective but not surjective.
It exists but may not be a function.
It is surjective but not injective.
A bijection is both one-to-one and onto, guaranteeing a two-sided inverse that also meets the criterion of a function. Thus f?¹ exists and assigns exactly one preimage to each element of B. Further reading.
For f: ? ? ? defined by f(x) = e?, what is the preimage of the interval (1, e²)?
(??, 2)
(?1, 1)
(1, e²)
(0, 2)
Solve e? ? (1,e²). Taking natural logs gives x ? (ln 1, ln e²) = (0,2). Therefore the preimage is (0,2). Further reading.
How many functions are there from a set of size 3 to a set of size 2?
9
8
12
6
Each of the 3 inputs has 2 possible outputs independently, so there are 2³ = 8 possible functions. This counts all mappings, not just injective ones. Further reading.
Let R be the relation on A = {1,2,3} defined by R = {(1,1),(2,2),(3,3),(1,2),(2,1)}. Which properties does R satisfy?
Only reflexive.
Reflexive and transitive only.
Reflexive and symmetric only.
Reflexive, symmetric, and transitive.
R contains (a,a) for all a, so it is reflexive. Since (1,2) and (2,1) are present, it is symmetric. All needed transitive pairs such as (1,2),(2,1)?(1,1) are also in R, making it transitive. Hence it is an equivalence relation. Further reading.
Which graphical test determines if a curve in the plane represents a function y = f(x)?
Vertical line test.
Intercept test.
Horizontal line test.
Rotation test.
The vertical line test states that a curve is the graph of a function if and only if no vertical line intersects the curve more than once. This ensures each x is paired with at most one y. Further reading.
Given the graph of a relation passes the vertical line test, what can you conclude?
It is a function.
It is one-to-one.
It is not a function.
It is onto.
Passing the vertical line test means no x-value maps to more than one y-value, which is the definition of a function. It does not address injectivity or surjectivity. Further reading.
If f: A ? B is a function, what is the kernel of f?
The set {(a1,a2): f(a1)=f(a2)}.
The set of all (a,a) for a in A.
The set of all b in B with preimages in A.
The set of a in A where f(a)=0.
The kernel of f is the equivalence relation on A defined by a1 ~ a2 if f(a1)=f(a2). This set of pairs measures exactly which inputs share the same output. Further reading.
Let f: ?\{2} ? ?\{?} be defined by f(x) = (3x + 1)/(x ? 2). What is f?¹(x)?
(2x + 1)/(x ? 3)
(2x ? 1)/(x ? 3)
(3x ? 1)/(x ? 2)
(x + 2)/(3 ? x)
To find the inverse, set y=(3x+1)/(x?2), solve for x: y(x?2)=3x+1 ? yx?2y=3x+1 ? x(y?3)=2y+1 ? x=(2y+1)/(y?3). Hence f?¹(x)=(2x+1)/(x?3). Further reading.
Consider the relation R on ? defined by aRb if 4 divides (a ? b). What is the equivalence class of 1?
{…, ?8, ?4, 0, 4, 8, …}
{…, ?6, ?2, 2, 6, 10, …}
{…, ?7, ?3, 1, 5, 9, …}
{…, ?5, ?1, 3, 7, 11, …}
The equivalence class of 1 mod 4 consists of all integers congruent to 1 modulo 4. Those are numbers of form 1+4k for k??, e.g. …,?7,?3,1,5,9,…. Further reading.
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Study Outcomes

  1. Define Functions and Relations -

    Understand formal definitions of relations and functions and recognize their distinguishing properties.

  2. Apply the Vertical Line Test -

    Use the vertical line test to quickly determine if a graph represents a function in practice relations and functions scenarios.

  3. Determine Domains and Ranges -

    Identify and list the domain and range from tables, graphs, and sets of ordered pairs accurately.

  4. Classify Functions in Various Representations -

    Decide which of the relations are functions by analyzing mappings, tables, and graphical data.

  5. Practice with Targeted Questions -

    Engage with multiple choice and open-ended questions for functions to reinforce your skills.

  6. Evaluate Relationships Independently -

    Assess new scenarios and determine is the relationship a function without step-by-step guidance.

Cheat Sheet

  1. Defining Relations vs. Functions -

    A relation is any set of ordered pairs, while a function assigns each domain element exactly one range element. Use the Vertical Line Test (VLT): if any vertical line crosses a graph more than once, the relation is not a function (Source: MIT OpenCourseWare). Keep practicing questions for functions by sketching quick graphs to spot non-functions instantly!

  2. Function Notation & Evaluation -

    Write functions as f(x), g(t), etc., to clearly identify input and output, then substitute to evaluate. For example, if f(x)=2x+3, then f(4)=2·4+3=11 (Source: Khan Academy). This notation makes it easier when you practice relations and functions problems - just plug in and solve!

  3. Domain & Range Fundamentals -

    The domain is all allowable inputs, and the range is all possible outputs for a function. For instance, g(x)=√(x−2) has domain x≥2 because under the radical you can't go negative (Source: University of California, Irvine). As you tackle questions for functions, list domain restrictions first to avoid common mistakes.

  4. One-to-One & Onto Functions -

    A one-to-one function pairs each output with at most one input, while onto covers every possible output value. Apply the Horizontal Line Test (HLT): if a horizontal line crosses more than once, it's not one-to-one (Source: Purplemath). Mnemonic: "HILO" helps you remember HLT checks if it's one-to-one.

  5. Composite Functions & Inverses -

    Composite functions combine two functions: (f∘g)(x)=f(g(x)). For example, if g(x)=x+1 and f(x)=3x, then (f∘g)(2)=f(3)=9 (Source: Coursera). To find an inverse f❻¹(x), swap x and y then solve - practicing "which of the relations are functions" often includes checking invertibility.

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