Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{5 x+1}{2-5 x} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the function

We are given the function: \( f(x) = \frac{5x + 1}{2 - 5x} \). Our goal is to find its inverse function, \( f^{-1}(x) \). This means we need to express \( x \) as a function of \( y \) and then solve for \( y \) in terms of \( x \).

Step 2: Set Up the Equation

We start by writing the function with \( y \) as the output. So, let \( y = \frac{5x + 1}{2 - 5x} \). The task is to solve this equation for \( x \) in terms of \( y \).

Step 3: Cross Multiply to Eliminate the Denominator

Multiply both sides by \( 2 - 5x \) to eliminate the fraction:\[y(2 - 5x) = 5x + 1\]This gives us: \[2y - 5xy = 5x + 1\]

Step 4: Rearrange the Equation

Reorganize terms to separate the \( x \) terms on one side:\[2y - 1 = 5x + 5xy\]Factor \( x \) from the right side:\[2y - 1 = x(5 + 5y)\]

Step 5: Solve for x

Divide both sides by \( (5 + 5y) \) to solve for \( x \):\[x = \frac{2y - 1}{5 + 5y}\]

Step 6: Write the Inverse Function

Since we set \( y = f(x) \) initially, the expression for \( x \) in terms of \( y \) becomes the inverse function:\[f^{-1}(x) = \frac{2x - 1}{5 + 5x}\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill used in solving equations and finding inverse functions. It involves rearranging algebraic expressions to isolate specific variables. Let's break down how this is applied when finding an inverse function.

When given a function, such as:

• \( f(x) = \frac{5x + 1}{2 - 5x} \)

Our first step is to alter it from \( y = f(x) \) to \( f^{-1}(x) \) by expressing \( x \) as a function of \( y \). To do this, use algebraic manipulation to move and rearrange terms.

• First, swap \( x \) and \( y \) in the equation, turning \( y \) into \( f \) and expressing it as: \( y = \frac{5x + 1}{2 - 5x} \).
• Next, manipulate the equation to solve for \( x \), using basic algebraic operations such as addition, subtraction, multiplication, and division.
• Systematically shift algebraic components around to isolate \( x \) on one side of the equation.

This rearrangement allows us to express \( x \) entirely in terms of \( y \) and ultimately reach the inverse function form.

Mastering algebraic manipulation is a crucial step in solving for inverse functions, as it ensures each variable is properly isolated and expressed accurately.

Solving Equations

Solving equations is at the heart of finding inverse functions. Once you've set up the initial function equation, you'll need to systematically solve for \( x \) as if it were the dependent variable.

Consider this equation derived from the function setting:

• \( y(2 - 5x) = 5x + 1 \)

Here, you need to:

• Distribute and eliminate terms to streamline the equation: \( 2y - 5xy = 5x + 1 \).
• Reorganize so that terms involving \( x \) are all on one side, leading to a simplified expression like \( x(5 + 5y) = 2y - 1 \).
• Factor or simplify where possible, reducing complexity and making the equation easier to solve.
• Finally, solve the equation by isolating \( x \): \( x = \frac{2y - 1}{5 + 5y} \).

Solving equations effectively requires practice in recognizing when and how to simplify, factor, or rearrange elements within an expression. By mastering these skills, inverse functions become more approachable and intuitive.

Cross Multiplication

Cross multiplication is an algebraic technique used to eliminate fractions from equations, simplifying the process of finding an inverse function.

When dealing with a function like:

• \( y = \frac{5x + 1}{2 - 5x} \)

Cross multiplication enables you to reorient the equation without a fraction:

• Multiply across, setting up as \( y(2 - 5x) = 5x + 1 \).
• By removing the fraction, the equation becomes straightforward to manipulate in the algebraic solving process.

Cross multiplication thus serves as the foundation allowing for clear algebraic manipulation by removing complex fractional expressions.

Practice cross multiplication in equations to efficiently manage transformations and ease the path toward finding inverse functions.