Problem 48 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=\frac{4 x-2}{3 x+1} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{-x - 2}{3x - 4} \).

Step 1: Set Equation for Inverse

The inverse function, denoted as \( f^{-1}(x) \), is obtained by swapping \( x \) and \( y \) in the original function and then solving for \( y \). Start by rewriting \( f(x) = y = \frac{4x - 2}{3x + 1} \) as \( x = \frac{4y - 2}{3y + 1} \).

Step 2: Clear the Denominator

Multiply both sides of the equation \( x = \frac{4y - 2}{3y + 1} \) by \( 3y + 1 \) to eliminate the fraction, resulting in \( x(3y + 1) = 4y - 2 \).

Step 3: Distribute and Rearrange Terms

Expand the left side to obtain \( 3xy + x = 4y - 2 \). Move all terms involving \( y \) to one side and constant terms to the other to get \( 3xy - 4y = -x - 2 \).

Step 4: Factor Out \( y \)

Factor \( y \) out from the left side: \( y(3x - 4) = -x - 2 \).

Step 5: Solve for \( y \)

Solve for \( y \) by dividing both sides by \( 3x - 4 \): \( y = \frac{-x - 2}{3x - 4} \). This gives the inverse function: \( f^{-1}(x) = \frac{-x - 2}{3x - 4} \).

Key Concepts

Algebra

Algebra is a central part of mathematics that helps us express general relationships between numbers and variables using symbols. It forms the foundation for understanding more complex mathematical concepts. In this exercise, algebra plays a key role in finding the inverse of a given rational function. To find the inverse function, we start by exchanging the positions of the dependent variable \( y \) and the independent variable \( x \). This process involves various algebraic manipulations to ultimately solve for the new dependent variable.

• We begin by setting the original function equal to \( y \), so \( y = \frac{4x - 2}{3x + 1} \).
• Next, we switch \( x \) and \( y \), resulting in \( x = \frac{4y - 2}{3y + 1} \).
• The algebraic task is to rearrange this new equation to solve for \( y \), which requires understanding and applying algebraic techniques like multiplication, addition, and factoring.

Algebra supports these manipulations by providing rules and properties that keep equations balanced throughout the operations.

Rational Functions

Rational functions are expressions that involve fractions where the numerator and the denominator are polynomials. In this exercise, the function \( f(x) = \frac{4x - 2}{3x + 1} \) is a rational function because both the numerator \(4x - 2\) and denominator \(3x + 1\) are linear polynomials.

• Key characteristics of rational functions include potential vertical asymptotes and horizontal asymptotes, which are important in describing the general behavior or shape of the function's graph.
• The original exercise involves determining the inverse of this rational function, a frequent application in various fields such as calculus and real-world problems.
• Special attention is needed during the transformation process, for example, ensuring that the expressions remain valid by not dividing by zero.

Understanding rational functions is crucial for recognizing the types of problems and solutions one can expect to encounter with them.

Function Operations

Function operations encompass a collection of processes such as addition, subtraction, multiplication, division, and finding inverses of functions. In this context, the operation of finding an inverse function is examined.

• Finding the inverse involves reversing the roles of the input and output, or more formally, the dependent and independent variables.
• The process includes isolating the new dependent variable, \( y \), which requires clear and logical step-by-step operations such as clearing fractions and factoring terms.
• Ensuring each operation respects algebraic properties is crucial, as this maintains the integrity and correctness of the result.

This focus on function operations allows us to explore how functions interact and transform, adding depth to our understanding of mathematical functions and their versatile applications.