Problem 70 - A Graphical Approach to Precalculus with Limits

Question

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{4-x}{5 x}$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{4}{5x + 1} \).

Step 1: Define the Inverse Function

To find the inverse of the function \( f(x) = \frac{4-x}{5x} \), we begin by setting \( y = f(x) \). This means \( y = \frac{4-x}{5x} \). We need to express \( x \) in terms of \( y \) to find \( f^{-1}(x) \).

Step 2: Solve for x in terms of y

Starting from \( y = \frac{4-x}{5x} \), multiply both sides by \( 5x \) to get rid of the fraction: \( 5xy = 4 - x \).

Step 3: Rearrange to Solve the Equation

Rearrange the equation \( 5xy = 4 - x \) by moving all terms involving \( x \) to one side: \( 5xy + x = 4 \).

Step 4: Factor Out x

From the equation \( 5xy + x = 4 \), factor out \( x \): \( x(5y + 1) = 4 \).

Step 5: Solve for x

Divide both sides by \( 5y + 1 \) to solve for \( x \): \( x = \frac{4}{5y + 1} \).

Step 6: Express x as f^-1(y)

Since \( x = f^{-1}(y) \), we can express this as \( f^{-1}(x) = \frac{4}{5x + 1} \) to find the inverse function.

Key Concepts

One-to-One Functions

Understanding the concept of a one-to-one function is crucial when finding inverse functions. A one-to-one function is a function where each output (or range value) corresponds to exactly one input (or domain value). This is also known as an injective function.
A one-to-one function has a very special property. Only such functions have inverses that are also functions. Why does this matter? Well, if multiple inputs map to the same output, you can't simply reverse the process, because it's unclear what the original input was. In the context of inverse functions, this unique pairing allows us to "undo" the function and find what input produced a given output.

• **Horizontal Line Test**: Graphically, you can verify a one-to-one function using the horizontal line test. If any horizontal line crosses the graph of the function more than once, then the function is not one-to-one.
• **Why one-to-one matters**: It ensures the inverse can also function ad a true mathematical function, where every input in its domain corresponds to one output.

When tackling exercises like finding the inverse function, recognizing whether the function is one-to-one is often your first step.

Function Composition

Function composition involves applying one function to the results of another. It's an essential tool when working with inverse functions, where you check your work by verifying if composing a function and its inverse results in the identity function.
Consider two functions, say, \( g(x) \) and \( f(x) \), where you define \( h(x) = f(g(x)) \). This is function composition. When finding the inverse of a function, \( f^{-1}(x) \) is said to "undo" \( f(x) \).

• **Identity Function**: This is a function where the output is the same as the input, typically expressed as \( I(x) = x \).
• **Verification by Composition**: If \( f(x) \) and its inverse \( f^{-1}(x) \) are correctly identified, then \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). These equalities confirm that the functions are truly inverses of each other.

Use function composition to test your inverse function results. It's a powerful tool to ensure accuracy in your solutions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and solving equations using mathematical operations like addition, subtraction, multiplication, division, and factoring.
Finding the inverse of a function often requires algebraic manipulation to express one variable in terms of another. When given a function, to find its inverse, you typically:

• **Start by swapping**: Write the function as \( y = f(x) \).
• **Solve for \( x\)**: Rearrange the equation to express \( x \) in terms of \( y \), as seen in the original solution.
• **Revert the variables**: Finally, express \( f^{-1}(x) \) such that \( y \) is written as \( x \) to get the inverse.

Engaging in algebraic manipulation is part of problem-solving that requires patience and practice. Follow logical steps and make sure to carefully track each operation to ensure correct inverse function identification.
By mastering these manipulations, you'll be equipped to tackle more challenging problems requiring a deep understanding of function inverses and their applications.