For the function $f(x)=\frac{3 x+2}{x}$, find $f^{-1}(x)$.

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02-11-2026
Mathematics

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For the function $f(x)=\frac{3 x+2}{x}$, find $f^{-1}(x)$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T15:51:18-05:00

Set the function equal to $y$ and swap the variables $x$ and $y$.
Solve the equation $x = \frac{3y+2}{y}$ for $y$ by multiplying through by $y$ and isolating the $y$ terms.
Factor $y$ from the expression $xy - 3y = 2$ to get $y(x - 3) = 2$.
Divide by $(x - 3)$ to obtain the inverse function: $\boxed{f^{-1}(x) = \frac{2}{x - 3}}$.

Explanation

Analysis of the Function and Goal

The goal is to find the inverse of the function $f(x) = \frac{3x+2}{x}$. An inverse function $f^{-1}(x)$ is a function that reverses the effect of $f(x)$. Specifically, if $f(a) = b$, then $f^{-1}(b) = a$. To find the inverse, we follow a systematic algebraic procedure. The function is defined for all $x eq 0$.

Setting up the Inverse Equation

To find the inverse, we start by setting $y = f(x)$, so we have:
\[y = \frac{3x+2}{x}\]
Next, we swap the roles of $x$ and $y$, which gives us the equation for the inverse relation:
\[x = \frac{3y+2}{y}\]

Solving for the Inverse Function

Now, we solve this equation for $y$ in terms of $x$.

Multiply both sides by $y$ to eliminate the denominator:

\[xy = 3y + 2\]

Move all terms containing $y$ to the left side by subtracting $3y$ from both sides:

\[xy - 3y = 2\]

Factor out $y$ on the left side:

\[y(x - 3) = 2\]

Finally, divide by $(x - 3)$ to isolate $y$:

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

Examples

Inverse functions are essential in various real-life scenarios where we need to reverse an operation. For instance, if a company has a pricing function where the cost $C$ of producing $x$ items is $C(x) = 5x + 100$, the inverse function $x(C) = \frac{C - 100}{5}$ allows the manager to determine how many items they can produce given a fixed budget $C$. Similarly, currency conversion rates work as functions, and finding the inverse allows you to quickly convert back to your home currency. Understanding these relationships helps in optimization, budgeting, and data analysis.