Problem 41 - Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. $$ f(x)=\frac{2}{x-3} $$

Step-by-Step Solution

Answer

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

Step 1: Set the Function Equal to y

Begin by setting the function equal to a new variable, typically \(y\): \[ y = \frac{2}{x-3} \]. This will help us in solving for \(x\) in terms of \(y\).

Step 2: Swap x and y

To find the inverse, swap \(x\) and \(y\) in the equation: \[ x = \frac{2}{y-3} \]. Now, you'll solve for \(y\).

Step 3: Clear the Denominator

Multiply both sides by \(y-3\) to clear the fraction: \[ x(y-3) = 2 \]. This simplifies the equation and makes it easier to solve for \(y\).

Step 4: Expand and Rearrange

Distribute \(x\) to both terms inside the parentheses: \[ xy - 3x = 2 \]. Then add \(3x\) to both sides to begin isolating \(y\): \[ xy = 2 + 3x \].

Step 5: Solve for y

Next, divide both sides of the equation by \(x\) to completely isolate \(y\): \[ y = \frac{2 + 3x}{x} \]. This gives the expression for the inverse in terms of \(x\).

Step 6: Express the Inverse Function

Since \(y\) represents the inverse function, express it in inverse function notation: \[ f^{-1}(x) = \frac{2 + 3x}{x} \]. This is the inverse function of the original function \(f(x)\).

Key Concepts

One-to-One Function

A one-to-one function, or injective function, means each output is paired with exactly one input. In simpler terms, no two different inputs give you the same output. This property is crucial for a function to have an inverse.
Consider this:

• If you have two outputs the same, then when you go in reverse, you won’t know where to go back to!
• So, a function must be one-to-one to ensure its inverse can "undo" it.
• Think of one-to-one like a locked door that only one person has the key to open; only they can claim they opened it.

Identifying a one-to-one function can involve the Horizontal Line Test in the graphing context: if no horizontal line intersects the graph of the function more than once, it is one-to-one.

Solving Equations

Solving equations is all about finding unknown values that make the equation true. Here's how solving contributes to finding an inverse function:

• First, rewrite the given function by replacing it with a variable like "y:" let's start with \(y = \frac{2}{x-3}\).
• Next, swap the roles of x and y to set the stage for finding the inverse, as in \(x = \frac{2}{y-3}\).
• Now, we solve for "y," which means making "y" the subject of the formula by performing operations step-by-step.
• Clear fractions by multiplying both sides by the denominator, rearrange terms, and finally isolate "y."

These steps mirror how we resolve algebraic expressions, ensuring we have a clear path from input to output in our inverse function.

Function Notation

Function notation is a way to denote functions and their inverses. Writing function notation like \(f(x)\) not only signals we have a function involving variable \(x\), but tells us what operations or transforms are happening.
Why does this matter?

• Function notation lets us uniquely identify functions and inverses, using symbols like \(f^{-1}(x)\) for an inverse function.
• This notation clarifies communication about functions and serves as a blueprint for calculations.
• In our problem, the inverse \(f^{-1}(x) = \frac{2 + 3x}{x}\) uses this notation to pinpoint exactly which operations "undo" the original transformation done by \(f(x)\).

Function notation thus acts not only as a label, but as a tool for precision and clarity in mathematics.