Problem 58 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1-x}{3 x+1} $$

Step-by-Step Solution

Answer

The inverse is the same as the original: \( f^{-1}(x) = \frac{1-x}{3x+1} \).

Step 1: Set the Function Equal to y

Set the function equal to a new variable, usually denoted as y, to make it easier to solve for the inverse. So we start with: \[ y = \frac{1-x}{3x+1} \]

Step 2: Swap x and y

To find the inverse function, swap x and y in the equation. This gives us:\[ x = \frac{1-y}{3y+1} \]

Step 3: Solve for y

The objective is to solve for y in terms of x. Begin by cross-multiplying to clear the fraction:\[ x(3y + 1) = 1 - y \] Expand the left side:\[ 3xy + x = 1 - y \] Rearrange terms to isolate y:\[ 3xy + y = 1 - x \] Factor out y:\[ y(3x + 1) = 1 - x \] Divide both sides by \(3x + 1\) to solve for y:\[ y = \frac{1-x}{3x+1} \]

Step 4: Conclude the Inverse Representation

After solving for y, we see that the original function and its inverse function are the same. Conclude:\[ f^{-1}(x) = \frac{1-x}{3x+1} \]

Key Concepts

Understanding Rational Functions

Rational functions are expressions that involve the quotient of two polynomials. A simple rational function is given in the form \[ f(x) = \frac{P(x)}{Q(x)} \]where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). The behavior of rational functions is interesting because their graphs can show vertical or horizontal asymptotes, or even holes, depending on the polynomials involved.In the given exercise, the function is\[ f(x) = \frac{1-x}{3x+1} \]which results from a simple linear polynomial in both the numerator and the denominator. The properties of this rational function are apparent in how the numerator and denominator behave. If the denominator approaches zero, the function could potentially become undefined, creating what is known as a vertical asymptote at those values of \( x \). To find vertical asymptotes, solve the expression \( 3x + 1 = 0 \). Horizontal asymptotes often occur when the degrees of the polynomials are equal, which leads to the asymptote being \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \). In this case, however, both coefficients are unified as 1, leading to no horizontal asymptote and a simpler function behavior.Understanding this foundation makes it easier to tackle problems involving inverse rational functions. By recognizing these patterns, you can effectively manipulate and transform the function.

Exploring Function Transformation

Function transformation is a method of adjusting the position and shape of a function's graph through certain modifications. These transformations include translations, reflections, and stretches or compressions.For transformations involving rational functions like \[ f(x) = \frac{1-x}{3x+1} \]transformations can affect both the numerator and denominator, impacting the overall behavior. When finding the inverse of a function, swapping variables is a unique transformation, which gives you a new perspective on the original graph. By replacing \( x \) with \( y \), you essentially flip the function over the line \( y = x \).Let's explore the transformation in our exercise:

• Swap Variables: The function becomes \( y = \frac{1-x}{3x+1} \) when set equal to \( y \).
• Reflection: Reflecting across the line \( y = x \) is accomplished by interchanging \( x \) and \( y \) (transform the graph's coordinates).
• Solve for the new variable: This involves algebraic manipulation to restore the function into a recognizable form.

Recognizing these transformation steps helps one grasp how to move from a function to its inverse and back, showcasing the graph's symmetry.

Mastering Algebraic Manipulation

Algebraic manipulation is crucial when working with functions, especially for finding inverses. This process involves rearranging and solving equations to arrive at a desired expression. To find an inverse, algebraic manipulation includes both basic arithmetic operations and factorization.Let's delve into the solution process for the given rational function:\[ x = \frac{1-y}{3y+1} \]Follow these precise steps:

• Cross-Multiply: Multiply each side by both denominators to eliminate fractions. This results in an equivalent equation without fractions.
• Expand and Rearrange: Distribute multiplication across summed or subtracted terms and move them around to isolate terms involving \( y \).
• Factor Out Variable: Organize the rearranged equation to factor out \( y \), simplifying the process of isolating \( y \).
• Simplify Further: Divide the coefficients to express \( y \) explicitly as a function of \( x \).
• Conclude: After thorough manipulation, the inverse is confirmed to be identical to the original function.

Algebraic manipulation is not just a routine procedure but an insightful way to polish your understanding of function inverses. The process reveals how careful operations can yield elegant solutions.