Inverse of a Function: Step-by-Step Guide, Rules & Solved Examples

The inverse of a function reverses the operation of the original function. If a function ${f}$ maps ${x}$ to ${y}$, the inverse function, denoted ${f^{-1}}$, maps ${y}$ back to ${x}$. A function must be one-to-one (bijective) to have an inverse, meaning each output has a unique input, and its graph is symmetrical along the line ${y=x}$.

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1. Introduction to the Inverse of a Function:

Understanding the inverse of a function is essential in advanced mathematics, especially in calculus and algebra. The inverse function essentially "reverses" the effect of the original function. For example, if a function transforms an input into an output, the inverse function returns the output to the original input. This concept is crucial in solving equations, analyzing graphs, and understanding real-life applications.

2. What is the Inverse of a Function:

An inverse function is a function that undoes the action of the original function. If a function ${f}$ maps ${x}$ to ${y}$, then the inverse function ${f^{-1}}$ maps ${y}$ back to ${x}$.

Mathematically, if: ${f(x)=y}$

Then the inverse is: ${f^{-1}(y)=x}$

For a function to have an inverse, it must be bijective. It must be both one-to-one (injective) and onto (surjective). This means every output value has a unique input value, and every possible output must be covered.

3. How to Find the Inverse of a Function:

To find the inverse of a function, follow these steps:

1. Replace ${f(x)}$ with ${y}$:
   Begin by rewriting the function in terms of ${y}$. For example, if ${f(x)=2x 3}$, write: ${y=2x 3}$

2. Swap ${x}$ and ${y}$:
   To find the inverse, interchange the variables: ${x=2y 3}$

3. Solve for ${y}$:
   Now, solve for ${y}$: ${y = \frac{x-3}{2}}$

4. Rewrite the inverse function:
   The inverse of ${f(x)}$ is: ${f^{-1}(x)= \frac{x-3}{2}}$

4. Rules for Inverse of a Function:

Certain rules must be followed when dealing with inverse functions:

• One-to-One Requirement: A function must be one-to-one to have an inverse, meaning no two different inputs should map to the same output.
• Horizontal Line Test: A function passes the horizontal line test if no horizontal line intersects its graph more than once. This indicates the function is one-to-one and has an inverse.
• Symmetry: The graph of a function and its inverse are symmetrical along the line ${y=x}$. Inverse functions reflect across this line.

5. Properties of Inverse of a Function:

Inverse functions have several important properties:

• Composition Property: If ${f}$ and ${f^{-1}}$ are inverses, then: ${f(f^{-1}(x))=x}$ and ${f^{-1}(f(x))=x}$

• Domain and Range: The original function's domain becomes the inverse function's range, and vice versa.

• Inverse of Linear Functions: For a linear function ${f(x)=ax b}$, the inverse will be another linear function ${f^{-1}(x)= \frac{x-b}{a}}$, provided ${a \neq 0}$.

• Inverse of Exponential and Logarithmic Functions: Exponential and logarithmic functions are inverses. For example: ${f(x)=e^x}$ and ${f^{-1}{(x)} =}$ In ${(x)}$

6. The inverse of a Function Solved Examples:

Question: 1
Find the inverse of ${f(x)=3x 7}$

Solution:

Step 1: Replace ${f(x)}$ with ${y}$: $\quad {y=3x 7}$

Step 2: Swap ${x}$ and ${y}$: $\quad \quad \quad \space \space{x=3y 7}$

Step 3: Solve for ${y}$: $\quad \quad \quad \quad \space \space \space \begin{matrix} x-7 = 3y \\ {y= \dfrac{x-7}{3}} \end{matrix}$

Final Inverse Function: $\quad \quad \space \space {f^{-1}(x)= \dfrac{x-7}{3}}$

Question: 2
Find the inverse of ${f(x)= \dfrac{2x-5}{3}}$

Solution:

Step 1: Replace ${f(x)}$ with ${y}$: $\quad {y= \dfrac{2x-5}{3}}$

Step 2: Swap ${x}$ and ${y}$: $\quad \quad \quad \space \space{x=\dfrac{2y-5}{3}}$

Step 3: Solve for ${y}$:
Multiply both sides by ${3}$: $\quad \quad \quad {3x = 2y - 5}$

Add ${5}$ to both sides: $\quad \quad \quad \quad \space {3x 5 = 2y }$

Divide by ${2}$: $\quad \quad \quad \quad \quad \quad \quad \space{y=\dfrac{3x 5}{2}}$

Final Inverse Function: $\quad {f^{-1}(x)= \dfrac{3x 5}{2}}$

Question: 3
Find the inverse of ${f(x)= 2x^3 1}$

Solution:

Step 1: Replace ${f(x)}$ with ${y}$: $\quad \quad \quad \quad {y = 2x^3 1}$

Step 2: Swap ${x}$ and ${y}$: $\quad \quad \quad \quad \quad \quad \space \space{x = 2y^3 1}$

Step 3: Solve for ${y}$:
Subtract ${1}$ from both sides: $\quad \quad \quad \quad \quad{x - 1 = 2y^3}$

Divide by ${2}$: $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \space \space{\dfrac{x-1}{2} = y^3}$

Take the cube root of both sides: ${\quad \quad \quad y = \sqrt[3]{\dfrac{x-1}{2}}}$

Final Inverse Function: $\quad \quad \quad \quad \quad {f^{-1}(x)= \sqrt[3]{\dfrac{x-1}{2}}}$

Question: 4
Find the inverse of ${f(x)= e^x}$

Solution:

Step 1: Replace ${f(x)}$ with ${y}$: $\quad \quad \quad \quad {y = e^x}$

Step 2: Swap ${x}$ and ${y}$: $\quad \quad \quad \quad \quad \quad \space \space{x = e^y}$

Step 3: Solve for ${y}$:
Take the natural logarithm of both sides: In${(x)=y}$

Final Inverse Function: $\quad \quad \quad \quad \quad {f^{-1}(x)=}$ In$(x)$

Question: 5
Find the inverse of ${f(x) =\dfrac{2x 1}{5}}$

Solution:

Step 1: Replace ${f(x)}$ with ${y}$: $\quad \quad \quad \quad {y = \dfrac{2x 1}{5}}$

Step 2: Swap ${x}$ and ${y}$: $\quad \quad \quad \quad \quad \quad \space \space{x = \dfrac{2y 1}{5}}$

Step 3: Solve for ${y}$:
Multiply both sides by ${5}$: ${\quad \quad \quad \quad \quad \quad 5x=2y 1}$

Subtract ${1}$ from both sides: ${\quad \quad \quad \quad \quad 5x-1=2y}$

Now, divide by ${2}$: ${\quad \quad \quad \quad \quad \quad \quad \quad \quad y = \dfrac{5x-1}{2}}$

Final Inverse Function: $\quad \quad \quad \quad \quad {f^{-1}(x)=\dfrac{5x-1}{2}}$

7. Practice Questions on Inverse of a Function:

Q.1 Find the inverse of the function ${f(x)=4x 7}$.

Q.2 Determine if the function ${f(x)=x^2}$ has an inverse.

Q.3 Find the inverse of ${f(x)= \frac{3x-1}{x 2}}$.

Q.4 Verify if ${f^{-1}(x)= \frac{x-4}{3}}$ is the inverse of ${f(x)=3x 4}$.

Q.5 Graph ${f(x)=x^2}$ and determine if it has an inverse.

8. FAQs on Inverse of a Function:

How do I know if a function has an inverse?

A function has an inverse if it is one-to-one, meaning it passes the horizontal line test.

What is the inverse of a quadratic function?

Not all quadratic functions have inverses because they are not one-to-one. However, restricting the domain can make some quadratic functions invertible.

What is the use of an inverse function?

Inverse functions are used to reverse mathematical processes, solve equations, and model real-world situations where an input needs to be retrieved from an output.

How can I graph an inverse function?

To graph an inverse function, reflect the original function's graph across the line ${y=x}$.

Why is the inverse of a function important?

Inverse functions allow us to "undo" the operation of a function, helping in algebra, calculus, and real-world applications like encryption, physics, and engineering.

Do all functions have inverses?

No, only one-to-one functions have inverses. Functions that map multiple inputs to the same output do not have inverses.

What is the relationship between a function and its inverse?

The function and its inverse are reflections of each other across the line ${y=x}$, and their compositions always return the input value.

9. Real-Life Application of Inverse of a Function:

Inverse functions have a variety of practical applications across different fields:

• Cryptography: Inverse functions are used in encryption algorithms to encode and decode messages.
• Physics and Engineering: Inverse functions help model systems where you must reverse a process, such as converting energy to force or time to velocity.
• Economics: Inverse functions can reverse supply-demand models or calculate inverse relationships between variables like price and quantity.

For example, in navigation, if you know the distance a vehicle traveled (output), using the inverse of the speed function helps find the time (input).

10. Conclusion:

The inverse of a function is a fundamental concept in mathematics, allowing us to reverse the effects of a function and solve various types of problems. It requires the function to be one-to-one and provides insight into the structure of mathematical models, equations, and real-life phenomena. Understanding how to find and work with inverses opens doors to advanced problem-solving techniques in algebra, calculus, and applied fields.

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