Problem 22 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Function Layout

The function given is \( f(x) = \frac{4x - 1}{2x + 3} \). This is a rational function where both the numerator and the denominator are linear expressions. Our task is to find a formula for the inverse function, which we'll denote as \( f^{-1}(x) \).

Step 2: Change \( f(x) \) to \( y \)

Start by replacing \( f(x) \) with \( y \). This gives us the equation \( y = \frac{4x - 1}{2x + 3} \). The goal is to solve this equation for \( x \), having \( y \) in terms of \( x \).

Step 3: Swap \( x \) and \( y \)

To find the inverse, interchange \( x \) and \( y \) in the equation. This results in \( x = \frac{4y - 1}{2y + 3} \). Now, we need to solve for \( y \) in terms of \( x \).

Step 4: Clear the Fraction

Multiply both sides by \( 2y + 3 \) to eliminate the fraction: \[ x(2y + 3) = 4y - 1 \]

Step 5: Distribute and Isolate \( y \)

Distribute \( x \) on the left side:\[ 2xy + 3x = 4y - 1 \]Then, move all terms involving \( y \) to one side, and constant terms to the other side:\[ 2xy - 4y = -1 - 3x \]

Step 6: Factor \( y \)

Factor out \( y \) from the left-hand side:\[ y(2x - 4) = -1 - 3x \]

Step 7: Solve for \( y \)

Divide both sides by \( (2x - 4) \) to solve for \( y \):\[ y = \frac{-1 - 3x}{2x - 4} \]

Step 8: Simplify the Expression

Notice that \( 2x - 4 \) can be simplified to \( 2(x - 2) \). Thus, the inverse function \( y \) simplifies to \[ y = \frac{-1 - 3x}{2(x - 2)} \].

Step 9: Write the Inverse Function

Therefore, the inverse function is \[ f^{-1}(x) = \frac{-1 - 3x}{2(x - 2)} \].

Key Concepts

Rational Function

A rational function is a ratio of two polynomials. In our exercise, the given function \( f(x) = \frac{4x - 1}{2x + 3} \) exemplifies a rational function. In this form, the numerator \( 4x - 1 \) and the denominator \( 2x + 3 \) are both linear expressions. These kinds of functions can represent a multitude of behaviors and are quite versatile, modeling anything from growth rates to decay curves. Rational functions can exhibit interesting features like vertical and horizontal asymptotes, which are values that the function approaches but never actually reaches. Understanding how the numerator and denominator interact can give great insights into the behavior of the function, especially when determining their domain and limits.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. To find the inverse of a function like \( f(x) = \frac{4x - 1}{2x + 3} \), you'll need to resolve it for one variable in terms of another. Start by expressing the function as \( y = \frac{4x - 1}{2x + 3} \). Then, swap the variables to seek the inverse: switch \( x \) and \( y \), leading to \( x = \frac{4y - 1}{2y + 3} \). This requires manipulating the equation to isolate \( y \), often involving steps such as clearing fractions by multiplication or distribution, and eventually rearranging terms until you have \( y \) isolated, yielding the inverse function. Techniques like factoring emerge essential throughout as they simplify complex expressions and make solving manageable.

Function Inversion

Function inversion is the process of reversing a function. It entails finding a new function \( f^{-1}(x) \) such that \( f(f^{-1}(x)) = x \). For the rational function \( f(x) = \frac{4x - 1}{2x + 3} \), the inverse function is derived through systematic steps: changing \( f(x) \, \text{to} \, y \), swapping \( x \) and \( y \), and then solving for \( y \). This process ultimately led us to the inverse \( f^{-1}(x) = \frac{-1 - 3x}{2(x - 2)} \). An important check is to substitute \( f^{-1}(x) \) back into \( f(x) \) and ensure it resolves to \( x \), confirming true inversion. Applying inversion holds significance as it reveals insights about the original function and provides solutions for equations set in a certain framework.

Linear Expressions

Linear expressions form the building blocks of the rational function in our example. A linear expression is a polynomial of the first degree, with one variable often expressed as \( ax + b \). In our function \( f(x) = \frac{4x - 1}{2x + 3} \), both the numerator and denominator are linear expressions: \( 4x - 1 \) and \( 2x + 3 \), respectively. These components directly influence the nature of the rational function, determining its slope, intercepts, and general behavior. Inverting a function that includes linear expressions involves comprehensive algebraic manipulation but offers the benefit of predictable, well-behaved results. Knowing how to handle and simplify these expressions eases the task of exploring more complex mathematical models, offering a solid foundation for advanced problem-solving.