📝 One to one functions and invertibility (12 MCQs)

📖 From Calculus • 1. Basics before calculus • 12 questions available

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What is One to one functions and invertibility?

Definition:
A one-to-one function (injective) maps distinct inputs to distinct outputs, i.e., $f(a) = f(b)$ implies $a = b$ , making it invertible over its range, with a unique inverse function defined everywhere on that range.

Example:
$f(x)=3x-1$ is one-to-one because if $3a-1=3b-1$ , then $a=b$ ; hence it is invertible, with inverse $f^{-1}(x) = (x+1)/3$ .

Reason:
One-to-one property is the key to invertibility, allowing for reversible processes in mathematics, cryptography, and data encoding.

Easy

Medium

Hard

📝 All One to one functions and invertibility MCQs

Q1. Which of the following best describes an increasing function on an interval?

A.For any

x_1<x_2

f(x_1)>f(x_2)

B.For any

x_1<x_2

f(x_1)<f(x_2)

. ✅

C.The function satisfies

f&#039;(x)=0

everywhere.

D.The function is symmetric about the y‑axis.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: An increasing function must satisfy the inequality $f(x_1)<f(x_2)$ whenever the input satisfies $x_1<x_2$ . Option B states exactly this condition, making it the correct definition. The other options either reverse the inequality or describe unrelated properties.

Q2. Suppose $f$ is strictly increasing on its domain and has an inverse $f^{-1}$ . Which of the following must be true for any $y_1<y_2$ in the range of $f$ ?

f^{-1}(y_1) > f^{-1}(y_2)

f^{-1}(y_1) = f^{-1}(y_2)

f^{-1}(y_1) < f^{-1}(y_2)

✅

D.No relationship can be deduced.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Because $f$ is strictly increasing, its inverse preserves order: smaller outputs correspond to smaller inputs. Hence if $y_1<y_2$ then the pre‑images satisfy $f^{-1}(y_1)<f^{-1}(y_2)$ . This logical consequence rules out the other choices.

Q3. Given two functions $f(x)=2x+3$ and $g(x)=2x-5$ , which statement correctly compares their monotonicity?

A.Both are decreasing because the coefficient of

x

is positive.

B.Both are increasing because the coefficient of

x

is positive. ✅

f

is increasing while

g

is decreasing.

D.Neither is monotonic.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The slope of each linear function is the coefficient of $x$ . Since both coefficients are $+2$ , each line rises as $x$ increases, making both functions strictly increasing. The other options misinterpret the sign of the slope or claim non‑monotonic behavior incorrectly.

Q4. If a function $h(x)=\sqrt{x}$ is defined for $x\ge0$ , what is its inverse $h^{-1}(x)$ ?

x^{2}

✅

\sqrt{x}

1/x

\ln x

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: To find the inverse, swap $y$ and $x$ : $x=\sqrt{y}$ implies $y=x^{2}$ . Because the original domain is $x\ge0$ , the inverse is defined for $x\ge0$ as $x^{2}$ . The other options either reproduce the original function or represent unrelated operations.

Q5. Consider a function $p$ that is strictly decreasing on $[-2,4]$ . Which of the following statements about its inverse $p^{-1}$ is necessarily true?

A.The graph of

p^{-1}

is a reflection of the graph of

p

across the line

y=x

. ✅

B.The graph of

p^{-1}

is a translation of the graph of

p

by 2 units up.

C.The inverse does not exist because decreasing functions fail the horizontal line test.

D.The inverse will be increasing.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When a function is one‑to‑one, its inverse exists and is obtained by reflecting each point $(a,b)$ to $(b,a)$ . This geometric operation is precisely a reflection across the line $y=x$ . The statement about translation or monotonic direction is incorrect; the correct relationship is the reflection.

Q6. Let $f(x)=x^{3}$ and $g(x)=\sqrt[3]{x}$ . Which of the following statements is accurate?

A.Both

f

and

g

are decreasing, and

g

is the inverse of

f

B.Only

f

is increasing;

g

is not invertible.

C.Neither function passes the horizontal line test.

D.Both

f

and

g

are increasing, and

g

is the inverse of

f

. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Both functions have positive derivative for all real $x$ , so each is strictly increasing. Moreover, $\sqrt[3]{x}$ undoes the cubing operation, making $g$ the exact inverse of $f$ . The other options either misstate monotonicity or ignore the inverse relationship.

Q7. Which point on the graph of $y = f(x) = 3x-2$ corresponds to the point $(5,13)$ on the graph of its inverse $f^{-1}$ ?

A.-135 ✅

B.-513

C.(13,-5)

D.513

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The inverse graph is obtained by swapping the coordinates of each point on the original graph. Therefore the point $(5,13)$ on $f^{-1}$ corresponds to $(13,5)$ on $f$ . This reflected point lies on the line $y=3x-2$ , confirming the answer.

Q8. Let $q(x)=\begin{cases} x+1 & x\le0 \\ 2x-3 & x>0 \end{cases}$ . Which of the following statements is true?

q

is strictly increasing on its entire domain, so it is invertible.

q

fails the horizontal line test because the two pieces intersect, so it is not invertible. ✅

q

is decreasing on

(-\infty,0]

and increasing on

(0,\infty)

, thus not one-to-one.

q

is strictly decreasing on its entire domain, so it is invertible.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The left branch yields values $\le1$ while the right branch starts at $-3$ and quickly exceeds those values, causing the same output to occur for different inputs (e.g., $-1$ and $1.5$ both give 0). Hence the function is not one‑to‑one and fails the horizontal line test.

Q9. If $f$ is strictly increasing and invertible, what is the result of the composition $f\bigl(f^{-1}(x)\bigr)$ for any $x$ in the range of $f$ ?

A.It equals

f(x)

B.It equals

f^{-1}(x)

C.It equals

x

. ✅

D.It is undefined.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: By definition of an inverse, applying $f$ after $f^{-1}$ returns the original argument: $f(f^{-1}(x))=x$ for every $x$ in the range of $f$ . This identity holds regardless of monotonicity, making option C the correct choice.

Q10. Let $h$ be defined implicitly by $x^{3}+y^{3}=6$ . On the interval where $y$ is a function of $x$ , which of the following statements is true about $h$ and its inverse?

h

is decreasing, and its inverse is also decreasing. ✅

h

is increasing, and its inverse is decreasing.

h

is decreasing, and its inverse is increasing.

h

is increasing, and its inverse is increasing.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Solving for $y$ gives $y=(6-x^{3})^{1/3}$ . Differentiating yields $dy/dx=-(x^{2})/(y^{2})$ , which is negative wherever defined, so $h$ is strictly decreasing. The inverse of a decreasing one‑to‑one function is also decreasing, confirming option A.

Q11. Consider $f(x)=\frac{1}{x}$ with domain $x\neq0$ . Which of the following correctly describes the relationship between the graph of $f$ and its inverse $f^{-1}$ ?

A.The graphs are identical because

f

equals its own inverse. ✅

B.The graph of

f^{-1}

is a vertical shift of the graph of

f

C.The graph of

f^{-1}

is a reflection of the graph of

f

across the x‑axis.

D.The graph of

f^{-1}

does not exist because

f

is not one‑to‑one.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For $f(x)=1/x$ , swapping $x$ and $y$ yields the same equation $y=1/x$ ; thus $f$ is its own inverse. Consequently, the two graphs coincide, making option A correct. The other statements mischaracterize the symmetry or claim non‑existence.

Q12. Which statement best synthesizes the connection between monotonicity, the horizontal line test, and invertibility for a continuous function $f$ on $\mathbb{R}$ ?

A.If

f

is monotonic (strictly increasing or decreasing), then every horizontal line intersects its graph at most once, guaranteeing the existence of a unique inverse function. ✅

B.Monotonicity guarantees a function passes the vertical line test, which ensures invertibility.

C.A function that fails the horizontal line test can still be invertible if it is piecewise monotonic.

D.The horizontal line test is irrelevant for continuous functions; only differentiability matters.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A strictly monotonic continuous function never takes the same value twice, so any horizontal line meets its graph at most once. This satisfies the horizontal line test, which is the precise criterion for the existence of a well‑defined inverse. The other options confuse different tests or ignore the key condition.

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