Problem 46 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=\frac{3 x}{x-2} $$

Step-by-Step Solution

Answer

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

Step 1: Replace f(x) with y

To begin finding the inverse of the function, replace \(f(x)\) with \(y\). Thus, the equation becomes:\[ y = \frac{3x}{x-2} \]

Step 2: Swap x and y

Next, interchange the roles of \(x\) and \(y\). This means we will write:\[ x = \frac{3y}{y-2} \]

Step 3: Solve for y

Now, solve the equation for \(y\). Start by eliminating the fraction:\[ x(y - 2) = 3y \]Expand the left side:\[ xy - 2x = 3y \]Rearrange the terms to isolate \(y\) terms on one side:\[ xy - 3y = 2x \]Factor out \(y\) on the left side:\[ y(x - 3) = 2x \]Divide by \((x - 3)\) to solve for \(y\):\[ y = \frac{2x}{x - 3} \]

Step 4: Finalize the inverse function

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

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating them. It is a unifying thread of almost all mathematics. When solving equations, as we do when finding the inverse of a function, algebra provides the rules that guide how we rearrange and simplify expressions.

Understanding how to manipulate variables and constants is fundamental in algebra. When we work through the steps of isolating a variable, such as solving for \( y \) in the inverse function exercise, algebra gives us tools like distributing terms, factoring, and isolating variables. This helps us express one variable explicitly in terms of another.

In the case of inverses, algebra allows us to manipulate the original function equation to find and express the inverse. This involves switching variables, rearranging, and performing operations such as addition, subtraction, multiplication, division, and factoring to isolate the needed variable.

Rational Functions

Rational functions consist of ratios of polynomials, where one polynomial is divided by another. They are written in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). In the given problem, the function \( f(x) = \frac{3x}{x-2} \) is an example of a rational function.

These functions can exhibit interesting behaviors, such as vertical asymptotes, which occur where the denominator equals zero. In our example, the rational function has a vertical asymptote at \( x = 2 \). This means \( f(x) \) will approach infinity as \( x \) approaches 2 from either side, making it undefined at \( x = 2 \).

When dealing with inverses of rational functions, it's important to find the domain of both the original function and its inverse. This is because the domain of the inverse will be the range of the original, except where the original function's operations disrupt continuity, like at asymptotes.

Function Notation

Function notation is a way to name and define functions in a concise, readable manner. It typically involves a letter like \( f \), followed by \( (x) \) to indicate it's a function of \( x \). For example, \( f(x) = \frac{3x}{x-2} \) clearly shows that \( f \) is a function dependent on \( x \), and provides the rule to compute the corresponding function values.

Using function notation makes it easy to work with and identify functions. It helps in finding an inverse function, denoted as \( f^{-1}(x) \). This notation quickly tells us that we have performed operations to reverse the effect of \( f(x) \).

Switching \( f(x) \) to \( y \), as in the problem, is a common step when finding an inverse. It simplifies visualization of variable manipulation and makes the algebraic solution more straightforward. In the end, translating \( y \) back to \( f^{-1}(x) \) confirms the inverse relationship between \( f \) and the solution found.