Problem 56 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to determine the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \frac{3x}{x-1} \). This means we must find a function that, when applied to \( f(x) \), results in the original input, \( x \).

Step 2: Swap Variables

To find the inverse, swap \( x \) and \( y \) in the function's equation. Start with \( y = \frac{3x}{x-1} \) and switch to \( x = \frac{3y}{y-1} \).

Step 3: Solve for y

Take the equation \( x = \frac{3y}{y-1} \) and solve for \( y \). Start by multiplying both sides by \( y-1 \): \( x(y-1) = 3y \).

Step 4: Distribute and Rearrange

Distribute \( x \) on the left side: \( xy - x = 3y \). Then, rearrange terms to get all terms involving \( y \) on one side: \( xy - 3y = x \).

Step 5: Factor y

Factor \( y \) out of the terms on the left: \( y(x - 3) = x \).

Step 6: Solve for y

Divide both sides by \( x - 3 \) to isolate \( y \): \( y = \frac{x}{x-3} \). This \( y \) represents \( f^{-1}(x) \).

Step 7: Verify the Inverse

Verify by checking if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Substitute \( f^{-1}(x) = \frac{x}{x-3} \) back into \( f(x) \) and vice versa. Both calculations should simplify to \( x \).

Key Concepts

Function Notation

In mathematics, function notation is a way of representing functions in a clear and systematic manner. Function notation uses symbols to define relationships between variables. For example, if you see the notation \( f(x) \), it indicates a function named \( f \) applied to an independent variable \( x \).
Here, \( f(x) = \frac{3x}{x-1} \) is a rational function, where \( x \) is the variable. This tells us how \( f \) behaves for different values of \( x \).

• The symbol \( f \) represents the function's name.
• \( x \) is the input to this function.
• \( f(x) \) is the output, or the result when \( x \) is processed through the function.

Function notation also helps with identifying properties like domain and range. Understanding these notations is crucial when working with inverse functions, as it is the notation that helps us see how functions map inputs to outputs and vice versa.

Solving Equations

Solving equations is a key process in finding inverse functions. When we talk about solving equations, we mean finding the values of the unknown variables that make the equation true. For the given function \( f(x) = \frac{3x}{x-1} \), to find the inverse function, we must solve the equation after swapping \( x \) and \( y \).
First, after swapping, we get the equation: \( x = \frac{3y}{y-1} \). The goal here is to solve for \( y \). This requires a step-by-step manipulation of the equation:

• Create an equation without fractions by multiplying through by \( y-1 \), giving \( x(y-1) = 3y \).
• Distribute \( x \) over \( y-1 \), resulting in: \( xy - x = 3y \).
• Rearrange the equation to collect all terms involving \( y \) on one side: \( xy - 3y = x \).
• Use algebraic manipulation to finally express \( y \) in terms of \( x \).

This new expression of \( y \) will be our inverse function \( f^{-1}(x) \), provided it satisfies the necessary conditions upon verification.

Algebraic Manipulation

Algebraic manipulation is essential in solving for inverse functions. This is the process of rearranging and simplifying equations using algebraic rules to isolate desired terms. In our example, to find the inverse of \( f(x) = \frac{3x}{x-1} \), we performed several algebraic manipulations.
Here’s a deeper look into this process:

• Distribute: Apply the distributive property to simplify terms. We distributed \( x \) across \( y-1 \) to eliminate the fraction.
• Rearrange: Move terms around to collect like terms together, usually all terms involving the variable of interest on one side.
• Factor: Extract common factors from terms to simplify the expression, in this case, factoring \( y \) out of \( xy - 3y \).
• Isolate: Solve for the variable of interest by performing inverse operations, such as dividing both sides by a term to isolate \( y \) in \( y(x - 3) = x \).

Once two sides of the equation are simplified, dividing by the coefficient of \( y \), like we did with \( x-3 \), isolates \( y \), giving the final inverse function, \( y = \frac{x}{x-3} \). Mastering these steps ensures accurate results.