Which Functions Are Invertible Select Each Correct Answer

Which Functions Are Invertible? A complete walkthrough to Identifying Invertible Functions

When studying functions, one of the most intriguing questions is: Which functions are invertible? Invertibility is a critical concept that appears across algebra, calculus, and many applied fields such as computer science and engineering. Understanding when a function has an inverse—and how to construct that inverse—helps you solve equations, model real‑world processes, and analyze data. This article walks through the key criteria for invertibility, offers practical examples, and provides a step‑by‑step method for selecting the correct answer when faced with multiple‑choice questions about invertible functions.

Introduction: The Essence of Invertibility

A function ( f: A \rightarrow B ) is invertible if there exists a function ( f^{-1}: B \rightarrow A ) such that:

Because of that, ( f^{-1}(f(x)) = x ) for every ( x \in A ) (left inverse), and
( f(f^{-1}(y)) = y ) for every ( y \in B ) (right inverse).

In simpler terms, applying the function and then its inverse (or vice versa) brings you back to where you started. Not every function has this property; the function must be bijective—both injective (one‑to‑one) and surjective (onto).

Injective: No two different inputs produce the same output.
Surjective: Every element in the codomain is the image of at least one element in the domain.

When both conditions hold, the function is a bijection, and its inverse exists and is unique.

Step 1: Check Injectivity

1.1 Horizontal Line Test (for real‑valued functions)

Graphically, a function ( f ) is injective if any horizontal line intersects its graph at most once. If a horizontal line cuts the curve twice or more, two distinct ( x )-values map to the same ( y )-value, breaking injectivity.

1.2 Algebraic Test

For a function defined by an algebraic expression, solve ( f(x_1) = f(x_2) ). If the only solution is ( x_1 = x_2 ), the function is injective. For example:

( f(x) = 3x + 5 ) → ( 3x_1 + 5 = 3x_2 + 5 ) → ( x_1 = x_2 ).
( f(x) = x^2 ) over ( \mathbb{R} ) → ( x_1^2 = x_2^2 ) → ( x_1 = \pm x_2 ). Two distinct solutions (unless restricted to non‑negative domain), so not injective over all reals.

1.3 Monotonicity

If a function is strictly increasing or strictly decreasing on its domain, it is automatically injective. To give you an idea, ( f(x) = e^x ) is strictly increasing, hence injective on ( \mathbb{R} ).

Step 2: Check Surjectivity

2.1 Range Equals Codomain

Determine the set of all possible outputs (range) of the function. If this set matches the codomain exactly, the function is surjective.

Example: ( f(x) = \sin x ) with domain ( \mathbb{R} ) and codomain ( \mathbb{R} ) is not surjective because outputs are confined to ([-1, 1]).
Example: ( g(x) = \tan x ) with domain ( \mathbb{R} \setminus {\frac{\pi}{2} + k\pi} ) and codomain ( \mathbb{R} ) is surjective, as every real number is attained.

2.2 Algebraic Verification

Solve ( f(x) = y ) for ( x ) in terms of ( y ). If a solution exists for every ( y ) in the codomain, surjectivity holds.

Example: ( h(x) = x^3 ) → ( x = \sqrt[3]{y} ) is defined for all real ( y ), so ( h ) is surjective onto ( \mathbb{R} ).

Step 3: Combine the Tests

A function is invertible iff it passes both injectivity and surjectivity tests. Because of that, if either fails, the function does not have a global inverse. On the flip side, restrictions on the domain or codomain can sometimes restore bijectivity.

Practical Examples and Common Pitfalls

Common Pitfalls

Assuming all continuous functions are invertible: Continuity alone does not guarantee injectivity or surjectivity.
Ignoring domain restrictions: A function may be invertible only on a subset of its natural domain.
Confusing surjectivity with “onto the real numbers”: The codomain must be specified; a function can be surjective onto a smaller set.

Multiple‑Choice Strategy: Selecting the Correct Answer

When faced with a question like “Which of the following functions is invertible?”, follow this systematic approach:

Want to learn more? We recommend why does b vitamin make pee yellow and why is basa fish banned for further reading.

Identify the domain and codomain given in the question.
Test for injectivity using algebraic simplification or the horizontal line test.
Test for surjectivity by examining the range and comparing it to the codomain.
Check for bijectivity: If both tests succeed, the function is invertible.
Look for red flags:
- Non‑one‑to‑one behavior (e.g., even powers).
- Output range that does not match the codomain.
- Undefined points in the domain that affect surjectivity.

Example Question

Which of the following functions is invertible?
Practically speaking, > A) ( f(x)=x^2 ) with domain ( \mathbb{R} ) and codomain ( \mathbb{R} ). On the flip side, > B) ( g(x)=x^3 ) with domain ( \mathbb{R} ) and codomain ( \mathbb{R} ). > C) ( h(x)=\sin x ) with domain ( [-\frac{\pi}{2},\frac{\pi}{2}] ) and codomain ( [-1,1] ).
D) ( k(x)=e^x ) with domain ( \mathbb{R} ) and codomain ( (0,\infty) ).

Solution

A) Not injective (e.g., (f(2)=f(-2))).
B) Injective (strictly increasing) and surjective onto ( \mathbb{R} ).
C) Injective (strictly increasing) and surjective onto ([-1,1]).
D) Injective and surjective onto ((0,\infty)).

All options B, C, and D are invertible. So if the question asks for the invertible function, the answer would be B because it is the only one that maps onto the entire real line without restriction. That said, if multiple answers are allowed, list all that satisfy bijectivity.

Constructing the Inverse Function

Once you’ve confirmed a function is invertible, you can explicitly find its inverse:

Write the equation ( y = f(x) ).
Swap the roles of ( x ) and ( y ): ( x = f(y) ).
Solve for ( y ) in terms of ( x ).
Replace ( y ) with ( f^{-1}(x) ).

Example:
( f(x)=\frac{2x-3}{5} ).
Swap: ( x = \frac{2y-3}{5} ).
Solve: ( 5x = 2y - 3 ) → ( 2y = 5x + 3 ) → ( y = \frac{5x+3}{2} ).
Thus, ( f^{-1}(x)=\frac{5x+3}{2} ).

Frequently Asked Questions (FAQ)

Question	Answer
Can a function be invertible if it’s not continuous?Think about it:	Yes, over ( \mathbb{R} ). Even so,
**Is the inverse of ( f(x)=\tan x ) the same as the inverse of ( f(x)=\arctan x )? Example: ( f(x)=e^x ) maps ( \mathbb{R} ) to ( (0,\infty) ), not all of ( \mathbb{R} ).
**Do all polynomial functions of odd degree have inverses?
Can a function be injective but not surjective?As an example, ( f(x)=x^2 ) is invertible when restricted to ( [0,\infty) ). Plus, odd‑degree polynomials are strictly monotonic when leading coefficient is positive and have range ( \mathbb{R} ). That's why a piecewise linear function can be bijective even if it has jump discontinuities.	Yes. **
What if the function is invertible only on a subset of its domain? Worth adding:	Then it is invertible restricted to that subset. It is injective but not surjective onto ( \mathbb{R} ).

Conclusion: Mastering Invertibility

Identifying whether a function is invertible boils down to checking two fundamental properties: injectivity and surjectivity. By applying algebraic tests, visual tools like the horizontal line test, and understanding domain/codomain constraints, you can confidently determine bijectivity. Once confirmed, constructing the inverse becomes a straightforward algebraic manipulation.

Whether you’re tackling exam questions, solving real‑world modeling problems, or simply deepening your mathematical intuition, mastering invertibility equips you with a powerful tool for analysis and problem‑solving. Keep practicing with diverse function types, and soon the process of spotting bijections will feel intuitive and natural.