📝 Definition of a function in math (15 MCQs)

📖 From Calculus • 1. Basics before calculus • 15 questions available

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What is Definition of a function in math?

Definition:
A function is a mathematical relation that uniquely assigns each element from a set called the domain to exactly one element in a set called the codomain, ensuring that every input has a single output, which forms the foundation of mathematical modeling and analysis.

Example:
Given $f(x) = 2x + 3$ , for $x = 1$ , $f(1) = 2(1) + 3 = 5$ , so the input 1 maps to output 5 uniquely.

Reason:
This definition ensures predictability and consistency in mathematics, allowing us to model real-world relationships where one cause produces one effect, such as distance traveled over time at constant speed.

Easy

Medium

Hard

📝 All Definition of a function in math MCQs

Q1. What is the correct definition of a function?

A.A relation where each x is paired with at most one y

B.A relation where each y is paired with exactly one x

C.A relation where each x is paired with exactly one y ✅

D.A relation where each y may have multiple x

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The definition of a function requires that every element of the domain (the x‑values) is associated with exactly one element of the codomain (the y‑value). This matches option C, which states that each x determines exactly one y. Options A, B, and D either relax or reverse this requirement, so they are incorrect.

Q2. Given the table of Indianapolis 500 qualifying speeds, does the relation between year (t) and speed (S) define a function?

A.Yes, because each year maps to exactly one speed ✅

B.No, because some years have multiple speeds

C.Yes, because speeds repeat for different years

D.No, because the table is incomplete

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Each year appears only once in the table, and for that year there is a single speed value. Hence the mapping satisfies the definition of a function: one input (year) yields one output (speed). Repeating speed values for different years do not violate the definition, making option A correct.

Q3. A graph shows a curve that loops back over itself, intersecting a vertical line at two points. What does this imply about the relation depicted?

A.The relation is a function because each x has a y

B.The relation is not a function because some x correspond to two y’s ✅

C.The graph represents a one‑to‑one correspondence

D.The relation is a function only on intervals where it passes the vertical line test

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The vertical line test states that a curve represents a function only if any vertical line intersects it at most once. Since the graph loops and a vertical line meets it at two points, at least one x‑value has two y‑values, violating the function definition. Therefore the relation is not a function, making option B correct.

Q4. Which method of representing a function is most appropriate for a continuously varying quantity like temperature over time?

A.Algebraic formula

B.Verbal description

C.Numerical table

D.Geometric graph ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Continuous data benefit from a visual representation that captures smooth change, and a geometric graph does this effectively. While formulas can also describe continuous relationships, the graph directly shows how temperature varies with time, making option D the best choice. Tables are discrete, and verbal descriptions lack visual clarity.

Q5. For a data set listing the number of students in each class, which representation best preserves the discrete nature of the information?

A.Algebraic formula

B.Numerical table ✅

C.Geometric graph

D.Verbal description

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When data are inherently discrete—each class has a specific count—a numerical table displays each class and its exact student count without interpolation. A formula would imply a continuous relationship, and a graph might suggest continuity between points. Therefore option B correctly captures the discrete character of the data.

Q6. Which of the following relations fails to be a function?

A.{(1,2),(1,3),(2,4)} ✅

B.{(0,5),(1,6),(2,7)}

C.{(3,9),(4,10),(5,11)}

D.{(6,12),(7,13),(8,14)}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A relation is a function only if each x‑value appears once. In set A, the x‑value 1 appears with two different y‑values (2 and 3), violating the definition. The other sets assign a single y to each distinct x, so they satisfy the function criteria. Hence option A is the correct answer.

Q7. A thermometer records temperature T as a function of time t throughout a day. Is this relationship a function?

A.No, because temperature can repeat values

B.Yes, because each moment in time yields a single temperature reading ✅

C.No, because time can be negative

D.Yes, because temperature determines time

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For each specific time t, the thermometer provides exactly one temperature reading, satisfying the definition of a function (one input → one output). The fact that temperature values may repeat at different times does not affect the function status, so option B is correct.

Q8. Newton’s law of universal gravitation is stated verbally: “The gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance.” Which algebraic expression correctly captures this as a function of distance r?

F(r)=\dfrac{G\,m_1\,m_2}{r^{2}}

✅

F(r)=G\,m_1\,m_2\,r^{2}

F(r)=\dfrac{G\,m_1\,m_2}{r}

F(r)=G\,m_1\,m_2\,r

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The verbal description translates to “force equals a constant G times the product of the masses divided by the distance squared.” This is precisely the formula $F(r)=\frac{G m_1 m_2}{r^{2}}$ . Options B, C, and D misplace the distance term, making option A the correct representation.

Q9. In the gravitational law $F = \dfrac{G m_1 m_2}{r^{2}}$ , which variable is the independent variable when the masses are held constant?

F

G

r

✅

m_1

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: When the masses $m_1$ and $m_2$ and the constant $G$ are fixed, the only variable that can change is the distance $r$ . Treating $r$ as the input determines the output force $F$ . Hence $r$ is the independent variable, making option C correct.

Q10. Consider the piecewise definition $f(x)=\begin{cases}x^{2}, & x\ge 0\\-x, & x<0\end{cases}$ . Does this define a function?

A.No, because the two pieces overlap

B.Yes, because each x yields exactly one y ✅

C.No, because the rule is not expressed algebraically

D.Yes, but only for x≥0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The definition assigns a single formula to each portion of the domain and the two portions do not intersect; every real number x falls into exactly one case, producing a unique y. Thus the relation satisfies the function definition, making option B correct.

Q11. For the function defined by $f(x)=\sqrt{x-3}$ , which set correctly describes its domain?

A.All real numbers

B.All positive numbers

C.All numbers greater than 3

D.All numbers greater than or equal to 3 ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The square‑root function is defined only when its radicand is non‑negative. Setting $x-3\ge0$ yields $x\ge3$ . Therefore the domain consists of all real numbers that are at least 3, which corresponds to option D.

Q12. If $y = 3x + 2$ and x increases by 1 unit, how does y change?

A.y increases by 3 units ✅

B.y increases by 2 units

C.y decreases by 1 unit

D.y remains unchanged

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The equation is linear with slope 3, meaning that for each unit increase in x, y increases by the slope value, which is 3. Hence a one‑unit increase in x results in a three‑unit increase in y, making option A correct.

Q13. A graph fails the vertical line test because a vertical line intersects it at three points. Which statement about this graph is true?

A.It correctly represents a function

B.It cannot represent a function

C.It represents a one‑to‑one correspondence ✅

D.It represents a function only on intervals where the test passes

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: When a vertical line meets the graph at multiple points, the relation violates the definition of a function. However, the graph may still represent a function on restricted intervals where the vertical line test is satisfied. Therefore option C accurately reflects this nuance.

Q14. Why does a verbal description like “speed is distance divided by time” qualify as a function of distance?

A.Because it uses words instead of symbols

B.Because each distance value yields one unique speed value ✅

C.Because it does not involve a graph

D.Because it cannot be expressed algebraically

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The verbal statement defines a rule that assigns to every distance a single speed value (speed = distance/time). This one‑to‑one mapping satisfies the function definition, even though it is expressed in words rather than symbols. Hence option B correctly explains why the description defines a function.

Q15. Given a table of years and speeds, a formula $S(t)=0.5t+200$ , and a graph derived from that formula, which representation would you use to predict the qualifying speed for the year 2012?

A.The table, because it contains the exact value

B.The formula, because it can be evaluated for any year ✅

C.The graph, because it visually shows the trend

D.All of the above, because they give the same prediction

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The formula provides an explicit expression that can be evaluated for any year, including 2012, regardless of whether the table includes that year. While the graph visualizes the trend, the formula yields the precise numerical prediction directly, making option B the most reliable choice.

🔗 Related Topics (MCQs)

📝 Independent and dependent variables in functions MCQs 📝 How to graph functions MCQs 📝 Vertical line test for functions MCQs 📝 Absolute value function properties MCQs 📝 Piecewise defined functions examples MCQs

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