Finding the Inverse of a One–One Function

Introduction

Key Concepts

Understanding Functions and Inverses

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Formally, a function $f$ from set $A$ to set $B$ is denoted as $f: A \rightarrow B$. For a function to possess an inverse, it must be one–one (injective). A one–one function ensures that each output is associated with a unique input, making the inverse function well-defined.

Definition of Inverse Function

The inverse of a function $f$, denoted as $f^{-1}$, reverses the mapping of $f$. If $f: A \rightarrow B$, then $f^{-1}: B \rightarrow A$ such that:

For a function to have an inverse, it must satisfy the horizontal line test, ensuring that no horizontal line intersects the graph of the function more than once.

Finding the Inverse of a One–One Function

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

1. Express the function in the form: $y = f(x)$
2. Swap $x$ and $y$: $x = f(y)$
3. Solve for $y$: Manipulate the equation to express $y$ in terms of $x$.
4. Replace $y$ with $f^{-1}(x)$: The resulting equation represents the inverse function.

For example, consider the function $f(x) = 2x + 3$. To find its inverse:

1. Start with $y = 2x + 3$.
2. Swap variables: $x = 2y + 3$.
3. Solve for $y$: $y = \frac{x - 3}{2}$.
4. Thus, $f^{-1}(x) = \frac{x - 3}{2}$.

Verifying the Inverse Function

Verification ensures that $f$ and $f^{-1}$ are true inverses. This can be done by composing the functions and checking if the result is the identity function:

Using the previous example:

Since both compositions yield $x$, the inverse function is verified.

Graphical Interpretation

The graph of a function and its inverse are reflections of each other across the line $y = x$. This property can be used to visually confirm the correctness of an inverse function. For example, plotting $f(x) = 2x + 3$ and its inverse $f^{-1}(x) = \frac{x - 3}{2}$ will show symmetry about the line $y = x$.

Examples of Inverse Functions

• Linear Functions: $f(x) = ax + b$ where $a \neq 0$.
• Quadratic Functions: Only one branch (e.g., $f(x) = x^2$ restricted to $x \geq 0$).
• Exponential Functions: $f(x) = e^x$ and its inverse $f^{-1}(x) = \ln(x)$.
• Logarithmic Functions: The inverse of an exponential function.

Applications of Inverse Functions

Inverse functions are essential in solving equations where the variable is inside a function. They are widely used in calculus, especially in integration and differentiation processes. Additionally, inverse functions play a critical role in various real-world applications such as cryptography, engineering design, and data analysis.

Common Mistakes to Avoid

• Assuming a function has an inverse without verifying if it is one–one.
• Incorrectly swapping variables or failing to solve for the correct variable.
• Neglecting to restrict the domain for functions that are not inherently one–one.

Practice Problems

1. Find the inverse of the function $f(x) = 3x - 5$.
2. Determine if the function $f(x) = x^3 + x$ has an inverse. Justify your answer.
3. Given $f(x) = \frac{2}{x + 1}$, find $f^{-1}(x)$.
4. Verify that your inverse function from problem 3 is correct by composing $f$ and $f^{-1}$.

Solutions to Practice Problems

1. To find $f^{-1}(x)$ for $f(x) = 3x - 5$:
   1. Start with $y = 3x - 5$.
   2. Swap variables: $x = 3y - 5$.
   3. Solve for $y$: $y = \frac{x + 5}{3}$.
   4. Thus, $f^{-1}(x) = \frac{x + 5}{3}$.
2. The function $f(x) = x^3 + x$ is one–one because its derivative $f'(x) = 3x^2 + 1$ is always positive, implying the function is strictly increasing. Therefore, it has an inverse.
3. To find $f^{-1}(x)$ for $f(x) = \frac{2}{x + 1}$:
   1. Start with $y = \frac{2}{x + 1}$.
   2. Swap variables: $x = \frac{2}{y + 1}$.
   3. Solve for $y$: $xy + x = 2$ → $xy = 2 - x$ → $y = \frac{2 - x}{x}$.
   4. Thus, $f^{-1}(x) = \frac{2 - x}{x}$.
4. Verification: $$ f(f^{-1}(x)) = f\left(\frac{2 - x}{x}\right) = \frac{2}{\left(\frac{2 - x}{x}\right) + 1} = \frac{2}{\frac{2 - x + x}{x}} = \frac{2}{\frac{2}{x}} = x $$ $$ f^{-1}(f(x)) = f^{-1}\left(\frac{2}{x + 1}\right) = \frac{2 - \frac{2}{x + 1}}{\frac{2}{x + 1}} = \frac{\frac{2(x + 1) - 2}{x + 1}}{\frac{2}{x + 1}} = \frac{2x}{2} = x $$ Both compositions yield $x$, confirming the inverse.

Advanced Concepts

Theoretical Foundations of Inverse Functions

Inverse functions are deeply rooted in the concept of bijections in set theory. A bijection is a function that is both injective (one–one) and surjective (onto), ensuring a perfect pairing between elements of the domain and codomain. The existence of an inverse function is guaranteed if and only if the original function is a bijection. This aligns with the principle that each element in the codomain must be mapped to by exactly one element in the domain.

Mathematical Derivations and Proofs

Proving that a given function has an inverse involves demonstrating that it is one–one. Consider the function $f(x) = x^3 + x$. To prove it is one–one:

1. Find the derivative: $f'(x) = 3x^2 + 1$.
2. Since $f'(x) > 0$ for all real $x$, the function is strictly increasing.
3. Therefore, it is one–one and has an inverse.

Another example involves proving the inverse of a trigonometric function within its principal domain. For instance, the function $f(x) = \sin(x)$ restricted to $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ is one–one, and its inverse is $f^{-1}(x) = \arcsin(x)$.

Complex Problem-Solving

Consider the function $f(x) = \frac{3x - 7}{2x + 5}$. Finding its inverse involves:

1. Start with $y = \frac{3x - 7}{2x + 5}$.
2. Swap variables: $x = \frac{3y - 7}{2y + 5}$.
3. Multiply both sides by $(2y + 5)$: $x(2y + 5) = 3y - 7$.
4. Expand and collect like terms: $2xy + 5x = 3y - 7$ → $2xy - 3y = -5x - 7$.
5. Factor out $y$: $y(2x - 3) = -5x - 7$.
6. Solve for $y$: $y = \frac{-5x - 7}{2x - 3}$.
7. Thus, $f^{-1}(x) = \frac{-5x - 7}{2x - 3}$.

Verification can be performed by composing $f$ and $f^{-1}$.

Interdisciplinary Connections

Inverse functions find applications across various disciplines:

• Physics: Inverse functions are used in kinematics to find time as a function of position.
• Economics: They help in determining supply and demand functions from price data.
• Engineering: In control systems, inverse functions are used to design controllers that achieve desired system behaviors.
• Computer Science: Cryptographic algorithms often rely on inverse functions for encoding and decoding messages.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$, are essential in solving equations involving trigonometric expressions. These functions provide angles corresponding to given trigonometric values within specific domains to maintain their one–one nature.

For example, to solve $\sin(\theta) = \frac{1}{2}$ for $\theta$, the inverse function $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$ provides the solution within the principal range.

Inverse Functions in Calculus

Inverse functions play a pivotal role in calculus, particularly in integration and differentiation:

• Derivative of Inverse Functions: If $y = f^{-1}(x)$, then the derivative is given by: $$ y' = \frac{1}{f'(f^{-1}(x))} $$
• Integration: Inverse functions are used in substitution methods for evaluating integrals.

For instance, if $f(x) = e^x$, then $f^{-1}(x) = \ln(x)$. The derivative of $\ln(x)$ is $\frac{1}{x}$, which aligns with the formula above.

Applications in Real-World Problem Solving

Inverse functions are instrumental in solving real-world problems where reversing a process is required. For example:

• Navigation: Determining the original speed of a vehicle given the distance and time.
• Finance: Calculating the principal amount based on future value and interest rates.
• Medicine: Reversing dosage calculations to determine the appropriate amount of medication based on patient weight.

These applications demonstrate the versatility and importance of understanding inverse functions in various contexts.

Challenges in Finding Inverse Functions

While finding the inverse of a one–one function is straightforward for linear functions, complexities arise with non-linear and higher-degree functions. Challenges include:

• Solving intricate equations where isolation of $y$ is non-trivial.
• Ensuring the function is one–one by restricting the domain.
• Handling functions with asymptotes or discontinuities that affect the existence of an inverse.

Advanced techniques, such as using logarithms or trigonometric identities, may be required to overcome these challenges.

Comparison Table

Summary and Key Takeaways