Sketching and Interpreting Results

Introduction

Key Concepts

Understanding Inverse Functions

An inverse function reverses the operation of the original function. If a function $f(x)$ maps an input $x$ to an output $y$, its inverse function, denoted as $f^{-1}(x)$, maps $y$ back to $x$. The existence of an inverse function requires that the original function be one-to-one, meaning it passes both the vertical and horizontal line tests.

Mathematically, a function $f(x)$ and its inverse satisfy the following conditions:

• $f(f^{-1}(x)) = x$
• $f^{-1}(f(x)) = x$

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

$y = 2x + 3$

$y - 3 = 2x$

$x = \frac{y - 3}{2}$

Thus, the inverse function is $f^{-1}(x) = \frac{x - 3}{2}$.

Graphical Interpretation of Inverse Functions

The graphs of a function and its inverse are mirror images across the line $y = x$. This symmetry is a key visual tool for sketching and interpreting the relationship between a function and its inverse.

To sketch the inverse of a function:

1. Graph the original function $f(x)$.
2. Draw the line $y = x$.
3. Reflect the graph of $f(x)$ over the line $y = x$ to obtain $f^{-1}(x)$.

For instance, if $f(x) = \sqrt{x}$, its inverse is $f^{-1}(x) = x^2$. Plotting both on the same axes with the line $y = x$ clearly shows their mirror symmetry.

Differentiation of Inverse Functions

Calculating the derivative of an inverse function is crucial in understanding its rate of change. If $y = f^{-1}(x)$, then its derivative can be expressed as:

Where $f'(x)$ is the derivative of the original function. This formula is derived using implicit differentiation.

**Example:** Find the derivative of the inverse function of $f(x) = e^x$.

First, find the inverse function: $f^{-1}(x) = \ln(x)$.

Then, compute the derivative using the formula:

$\frac{d}{dx} f^{-1}(x) = \frac{1}{f'\left(f^{-1}(x)\right)} = \frac{1}{e^{\ln(x)}} = \frac{1}{x}$

Applications of Inverse Functions

Inverse functions are widely used in various applications, including:

• Solving Equations: Inverse functions can simplify the process of solving equations by isolating variables.
• Real-World Modeling: They model real-world phenomena where reversing processes is necessary, such as converting between different unit systems.
• Optimization Problems: Inverse functions help in finding maximum or minimum values in optimization scenarios.

**Example:** To find the time required for an investment to reach a certain value with continuous compounding interest, the inverse of the exponential function is used.

Composite and Implicit Inverse Functions

Inverse functions often appear in composite functions, where the composition of a function and its inverse simplifies to the identity function. In implicit differentiation, inverse functions play a role in differentiating equations not easily solvable for one variable in terms of another.

For example, given the equation $y \cdot e^y = x$, finding $y$ in terms of $x$ involves an implicit inverse function, and differentiating such equations requires applying the chain rule and the derivative of the inverse function.

Properties of Inverse Functions

Several key properties of inverse functions are essential for sketching and interpretation:

• One-to-One: A function must be one-to-one to have an inverse.
• Domain and Range: The domain of $f$ becomes the range of $f^{-1}$, and vice versa.
• Symmetry: As mentioned, the graphs are symmetric about the line $y = x$.
• Interchangeability: Composing a function with its inverse yields the identity function.

Example Problems

Working through examples reinforces understanding of inverse functions:

Example 1: Find the inverse of $f(x) = \frac{2x - 5}{3}$.

Set $y = \frac{2x - 5}{3}$.

$3y = 2x - 5$

$2x = 3y + 5$

$x = \frac{3y + 5}{2}$

Thus, $f^{-1}(x) = \frac{3x + 5}{2}$.

Example 2: If $f(x) = \tan(x)$, find $f^{-1}(x)$ and its derivative.

The inverse function is $f^{-1}(x) = \arctan(x)$.

Its derivative is:

$\frac{d}{dx} \arctan(x) = \frac{1}{1 + x^2}$

Comparison Table

Summary and Key Takeaways