Problem 69 - A Graphical Approach to Precalculus with Limits

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: Swap Function and Variable

Since we are finding the inverse of the function, we start by rewriting the function equation with the dependent \(y\) and independent \(x\) variables interchanged. The equation is \(y = \frac{1-2x}{3x}\). For the inverse, replace \(f(x)\) with \(y\) and \(x\) with \(y\), giving us \(x = \frac{1-2y}{3y}\).

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides by \(3y\): \(3xy = 1 - 2y\). This expression helps us to isolate \(y\).

Step 3: Rearrange Equation

Reorganize the expression to isolate terms with \(y\) on one side: \(3xy + 2y = 1\).

Step 4: Factor Out \\(y\\)

Factor \(y\) from the terms on the left: \(y(3x + 2) = 1\). This simplification helps us solve for \(y\).

Step 5: Solve for \\(y\\)

Divide both sides by \(3x + 2\) to isolate \(y\): \(y = \frac{1}{3x + 2}\). This is our inverse function: \(f^{-1}(x) = \frac{1}{3x + 2}\).

Key Concepts

One-to-One Functions

Understanding one-to-one functions is crucial for working with inverse functions. A one-to-one function is a special type of function where each element of the domain is paired with a unique element of the range. This means no two different input values produce the same output value.

This unique pairing is significant because it guarantees that the function has an inverse. If a function is not one-to-one, it may not be possible to reverse the pairing and define a function that "undoes" the original function.

• A simple way to test if a function is one-to-one is using the Horizontal Line Test. If any horizontal line crosses the graph of the function at most once, then the function is one-to-one.
• Knowing a function is one-to-one gives confidence that we can safely find and work with its inverse.

Solving Equations

Solving equations is a fundamental skill in finding inverse functions, as it involves manipulating an equation until we isolate one variable in terms of another. In finding the inverse, you start by swapping the dependent and independent variables, then solve for the new output variable.

Here’s the step-by-step process reflected in the example function:

• Start by replacing the function notation with algebraic variables, i.e., replace \(f(x)\) with \(y\).
• Switch the roles of \(x\) and \(y\) to begin finding the inverse, which essentially means writing the output as an input.
• Next, solve the resulting equation for \(y\) by performing algebraic operations to isolate \(y\).

This process transforms the function into its inverse form.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to isolate specific variables using operations like addition, subtraction, multiplication, division, and factorization.

In our example, the goal is to solve for \(y\) (which will become the inverse function). Here's a recap of critical algebraic steps:

• Start by eliminating fractions by multiplying both sides by a common denominator, simplifying the equation into a more workable form.
• Reorder terms to gather similar or related terms which help in simplifying and solving equations.
• Factor out necessary terms to isolate the variable of interest conveniently. This uses distribution in reverse.
• Finally, divide by remaining coefficients to completely isolate \(y\).

Mastering these steps in algebraic manipulation makes it easier to tackle problems involving inverse functions.