How to Find the Inverse of a Function (With Solved Example)
The concept of inverse functions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding inverse functions helps students solve equations, analyse graphs, and quickly convert values between related domains—skills essential for CBSE, JEE, and other competitive exams.
What Is Inverse Functions?
An inverse function is a function that exactly “reverses” the action of another function. In other words, if you apply a function f to a value and then apply its inverse f−1 to the result, you will get the original value back: f(f−1(x)) = x and f−1(f(x)) = x. You’ll find this concept applied in areas such as function verification, domain and range analysis, and even in real-life conversions and cryptography.
Key Formula for Inverse Functions
Here’s the standard formula:
If f(x) = y, then the inverse function f−1(y) = x
Or, algebraically: f−1(x) is found by replacing f(x) with y, swapping x and y, and then solving for y.
Cross-Disciplinary Usage
Inverse functions are not only useful in Maths but also play an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions that involve conversions, encryption, or system modelling. Many real-world formulas (like converting Celsius to Fahrenheit and vice versa) are based on the concept of function inverses.
Step-by-Step Illustration
Let’s see the process to find the inverse function algebraically using an example:
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Given: f(x) = 3x + 5
Replace f(x) with y: y = 3x + 5 - Swap x and y: x = 3y + 5
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Solve for y:
x − 5 = 3y
y = (x − 5) / 3 - Write the inverse: f−1(x) = (x − 5) / 3
Verification: Let’s check one value:
f(2) = 3 × 2 + 5 = 11
So, f−1(11) = (11 − 5) / 3 = 2
Thus, the inverse “undoes” the function.
Speed Trick or Vedic Shortcut
A fast way to check if a function has an inverse is to use the “one-to-one” property: If every input has a unique output (no horizontal line crosses the graph more than once), then the function is invertible. Visually, this is called the “horizontal line test.” In board or MCQ exams, remember that all linear functions (except horizontals) are always invertible.
Example (CBSE fast check): Is f(x) = x2 invertible for all real x?
No, because f(2) = f(−2) = 4 (not one-to-one). But if we restrict the domain, say x ≥ 0, then it is invertible!
Vedantu’s live classrooms often use graph slides to help students visually “see” the inverse relationship.
Try These Yourself
- Find the inverse of f(x) = 2x − 1.
- Is f(x) = 5 − x invertible? Find its inverse.
- If f(x) = x3, what is f−1(x)?
- For which domain is f(x) = x2 invertible?
- What’s the inverse of converting Celsius to Fahrenheit: F = (9/5)C + 32?
Frequent Errors and Misunderstandings
- Assuming all functions have inverses (only one-to-one functions do).
- Forgetting to swap x and y when finding the inverse.
- Mistaking the inverse function for a reciprocal (inverse is not always 1/f(x)).
- Not checking that domain and range swap in the inverse.
- Skipping the step of verifying by composing the functions.
Relation to Other Concepts
The idea of inverse functions connects closely with one-to-one functions, domain and range, composition of functions, and inverse trigonometric functions. Mastering inverses helps students solve advanced algebraic and calculus problems, and it builds a strong foundation for competitive exams like JEE and NEET.
Classroom Tip
A quick way to remember inverse functions: Imagine “rewinding” a mathematical process—the inverse function takes you backward. Graphically, the function and its inverse are mirror images across the line y = x. Teachers at Vedantu show this using colored graph paper for visual clarity in live classes.
Inverse Functions in Real Life & Exams
Inverse functions are used in everyday conversions—like changing temperature between Celsius and Fahrenheit, swapping coordinates in geometry, or decrypting information in computer science. Board and CBSE exams often ask for stepwise methods, properties, or real-life application of inverses. Aim to practice at least 3–5 solved questions for exam confidence.
Wrapping It All Up
We explored inverse functions—from their formal definition, formula, and worked examples, to mistakes and connections with other maths topics. Continue practicing inverse functions with Vedantu to become confident in exam scenarios and real-life applications alike!
Key Internal Links
- One-to-One Function: Why only one-to-one functions have inverses.
- Domain and Range: Understanding domain-range switch in inverses.
- Composition of Functions: Relates to inverse verification f(f−1(x)) = x.
- Inverse Trigonometric Functions: Direct link for advanced practice.
FAQs on Inverse Functions Explained: Steps, Properties & Practice
1. What is an inverse function and how is it different from a reciprocal?
An inverse function reverses the effect of the original function, mapping each output back to its input. If a function is represented as f(x), its inverse is shown as f⁻¹(x). A reciprocal is the multiplicative inverse (1/x) and is not always related to the function's input-output reversal. Inverse functions switch the x and y values, while reciprocals focus only on multiplying to give 1.
2. How do you determine if a function has an inverse?
A function has an inverse if and only if it is both one-to-one (injective) and onto (surjective). This means every output is matched with one unique input, and all possible outputs are covered. This is also known as a bijective function.
3. What are the steps to find the inverse of a given function?
To find the inverse function f⁻¹(x):
1. Replace f(x) with y
2. Swap x and y in the equation
3. Solve the resulting equation for y
4. Replace y with f⁻¹(x)
4. Can a non-bijective function have an inverse?
No, a function must be bijective (both one-to-one and onto) to have an inverse. If a function is not one-to-one, multiple inputs map to the same output, preventing a unique inverse.
5. Why is verifying the inverse important, and how can you do it efficiently?
Verifying the inverse ensures the correctness of your solution. You can do this by checking if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Substitute a value or symbolically show both compositions result in x for verification.
6. Give an example of a function and its inverse.
Example: If f(x) = 3x + 2, its inverse is found as follows: set y = 3x + 2, swap to get x = 3y + 2, then solve for y: y = (x - 2)/3. Thus, f⁻¹(x) = (x - 2)/3.
7. What conceptual traps should students avoid when finding inverses?
Common traps include:
• Not checking the one-to-one and onto nature before attempting the inverse
• Omitting the domain and range swap
• Incorrectly solving after swapping variables
8. How are inverse functions applicable in real-life contexts?
Inverse functions are used in decoding messages, reversing operations in technology (like encryption and decryption), and in everyday calculations such as converting units or undoing effects in finance or science.
9. What is the relationship between domain and range when considering inverse functions?
The domain of a function becomes the range of its inverse, and vice versa. This interchanging ensures that every input-output pair is correctly reversed.
10. How do you handle inverse functions in MCQs and one-mark questions?
For MCQs, quickly check if the function is linear (which is always invertible), focus on the method steps (swap and solve), and look for direct options matching solved inverses. Practice frequently asked types for improved accuracy.
11. What happens to the graph of a function when you find its inverse?
The graph of the inverse function is a reflection of the original function's graph across the line y = x. This is because the x and y coordinates are swapped.
12. How do I determine the domain and range of an inverse function?
The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function. Always consider any restrictions on the original function's domain to determine the range (and therefore the domain) of the inverse.
