Problem 69 - A Graphical Approach to College Algebra

Question

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{1-2 x}{3 x}$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{1}{3x + 2} \).

Step 1: Replace f(x) with y

Since we are dealing with the function, let's rewrite it in terms of y. \[ y = \frac{1 - 2x}{3x} \]

Step 2: Swap x and y

To find the inverse function, swap the variables x and y. \[ x = \frac{1 - 2y}{3y} \]

Step 3: Solve for y

To find the inverse function, solve the equation for y.Multiply both sides by 3y:\[ 3xy = 1 - 2y \]Rearrange to get all terms involving y on one side:\[ 3xy + 2y = 1 \]Factor y out:\[ y(3x + 2) = 1 \]Finally, solve for y by dividing both sides by \(3x + 2\):\[ y = \frac{1}{3x + 2} \]

Step 4: Write the inverse function

Now that we have expressed y in terms of x, we can write the inverse function as:\[ f^{-1}(x) = \frac{1}{3x + 2} \]

Key Concepts

One-to-One Function

Understanding the concept of a one-to-one function is crucial when dealing with inverse functions. A one-to-one function, often symbolized as "1-1", is a function where each output is mapped to exactly one input. This means that if you have two different inputs, they must always produce two different outputs. In more mathematical terms, for a function to be one-to-one, if \(f(a) = f(b)\) then it can be inferred that \(a = b\). Essentially, it means that every horizontal line should intersect the graph of the function at most once.
Some important features of one-to-one functions include:

• They always have an inverse that is also a function.
• They pass the horizontal line test.
• They are either strictly increasing or strictly decreasing.

A common example of a one-to-one function is a linear function with a non-zero slope, such as \(f(x) = 2x + 3\). This linear formula ensures that each "x" value produces a unique "y" value.

Algebraic Manipulation

Algebraic manipulation is the process we use to rearrange equations and expressions in mathematics. It allows us to isolate a desired variable, which is essential when finding inverse functions. Let's break down the process using our exercise: First, we start with the function expressed as \(y = \frac{1 - 2x}{3x}\). To find the inverse, we swap \(x\) and \(y\) to get \(x = \frac{1 - 2y}{3y}\).
Then, solve for \(y\) by:

• Multiplying through by \(3y\) to eliminate the denominator: \(3xy = 1 - 2y\).
• Rearranging to gather terms involving \(y\): \(3xy + 2y = 1\).
• Factoring out \(y\) from the left side: \(y(3x + 2) = 1\).
• Finally, divide by the bracketed term to isolate \(y\): \(y = \frac{1}{3x + 2}\).

Repeatedly, these manipulations help us express one variable as a function of another, a vital step in deriving inverses.

Function Notation

Function notation is a way to represent functions in algebra that makes them easy to understand and work with. It resembles this structure: \(f(x)\). The "\(f\)" is the function name, and "\(x\)" is the input variable. In the context of finding inverse functions, function notation helps indicate which transformation we are performing. For example, in the inverse function \(f^{-1}(x)\), the "\(-1\)" superscript denotes the inverse operation.
Here's how function notation plays a role:

• It helps distinguish the original function from its inverse, e.g., \(f(x)\) versus \(f^{-1}(x)\).
• It provides clear input-output pairing, simplifying analysis and calculation.
• The notation allows functions to be easily composed with other functions, which is common in complex mathematical operations.

Function notation thus offers an organized and consistent framework, enabling mathematicians and students to communicate and solve problems more effectively.