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Chapter 8: Problem 128

The function \(f(x)=\frac{3-x}{2 x-5}\) is one-to-one. Find \(f^{-1}\).

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{3 + 5x}{2x + 1} \).

Step by step solution

01

Interchange f(x) and y

First, let y be equal to the function: \( y = \frac{3-x}{2x-5} \). Our goal is to solve for x in terms of y.
02

Cross-multiply

Cross-multiplying to eliminate the fraction gives: \( y(2x - 5) = 3 - x \).
03

Distribute y

Distribute y on the left-hand side: \( 2xy - 5y = 3 - x \).
04

Collect x terms

Move all terms involving x to one side and constants to the other side: \( 2xy + x = 3 + 5y \).
05

Factor out x

Factor x out of the left-hand side: \( x(2y + 1) = 3 + 5y \).
06

Solve for x

Solve for x by isolating it: \( x = \frac{3 + 5y}{2y + 1} \).
07

Replace x with f^{-1}(x)

Replace y with x to obtain the inverse function: \( f^{-1}(x) = \frac{3 + 5x}{2x + 1} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding One-to-One Functions
A function is considered one-to-one if each input produces a unique output, meaning no two different inputs will map to the same output. This is crucial for determining if a function has an inverse because only one-to-one functions have inverses that are also functions.
To test if a function is one-to-one, you can use the Horizontal Line Test. If any horizontal line intersects the graph of the function at most once, the function is one-to-one. For the function given in the exercise,\( f(x) = \frac{3-x}{2x-5} \), this property holds. This means we can confidently find its inverse.
Cross-Multiplication to Eliminate Fractions
Cross-multiplication is a technique used to eliminate a fraction by rearranging an equation. It's extremely useful when solving rational equations, which involve fractions. In our exercise, the function \( y = \frac{3-x}{2x-5} \) includes a fraction, so we cross-multiply to simplify the equation.
By multiplying both sides by the denominator, we get:
    \[ y(2x - 5) = 3 - x \]
This transforms our equation into a more manageable form without the fraction, enabling easier manipulation in the next steps.
Applying Algebraic Manipulation
Algebraic manipulation involves a variety of techniques to simplify or solve equations. For our problem, we used several manipulations, including distributing, combining like terms, and factoring. After cross-multiplying, our equation was:
    \[ y(2x - 5) = 3 - x \]
We distributed \( y \) on the left side to get:
    \[ 2xy - 5y = 3 - x \]
Next, we moved all terms involving \( x \) to one side:
    \[ 2xy + x = 3 + 5y \]
Finally, we factored out \( x \) on the left side, resulting in:
    \[ x(2y + 1) = 3 + 5y \]
Each step brought us closer to isolating \( x \), which is our goal.
Solving for x in Terms of y
The ultimate goal when finding an inverse function is to solve for \( x \) in terms of \( y \). After algebraically manipulating the equation to bring all \( x \)-terms to one side and factor out \( x \), we got:
    \[ x(2y + 1) = 3 + 5y \]
We then isolated \( x \) by dividing both sides by \( (2y + 1) \), leading to:
    \[ x = \frac{3 + 5y}{2y + 1} \]
Finally, since we let \( y \) represent \( f(x) \) in the beginning, we replace \( y \) back with \( x \) to write the inverse function:
    \[ f^{-1}(x) = \frac{3 + 5x}{2x + 1} \]
This shows the steps to transition from our original function to its inverse, completing the problem.

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