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Chapter 6: Problem 21

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

Short Answer

Expert verified
The inverse of \( f(x) = -\frac{3}{x} \) is \( f^{-1}(x) = -\frac{3}{x} \). The function is its own inverse.

Step by step solution

01

Define the Equation for Inversion

Start by setting the function equal to a variable: let \( y = f(x) = -\frac{3}{x} \). This establishes \( y \) as the output of the function for a given \( x \).
02

Swap Variables for Inversion

To find the inverse, swap \( x \) and \( y \). This means we now have \( x = -\frac{3}{y} \). This step sets up the equation from which we'll solve for the inverse.
03

Solve for y

Isolate \( y \) by multiplying both sides of the equation by \( y \) to get \( xy = -3 \). Next, solve for \( y \) by dividing both sides by \( x \), resulting in \( y = -\frac{3}{x} \).
04

Confirm the Inverse

Notice that the function we derived, \( y = -\frac{3}{x} \), is identical to the original function \( f(x) \). This means that the function is its own inverse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Inversion
Finding the inverse of a function is like solving a mirror puzzle. You want to pinpoint what input (or "x") led to a particular output (or "y").

To start, you take the function you have and express it as an equation. Let's look at an example: we have the function \( f(x) = -\frac{3}{x} \). To find the inverse, we first write it as \( y = f(x) = -\frac{3}{x} \). This helps put us in the right direction for finding the inverse.
  • The concept of function inversion involves swapping the roles of "x" and "y."
  • Once swapped, the equation can be solved for "y," giving us the inverse function.
This swapping helps us see the function as a reversed process, where outputs can efficiently tell us the inputs.
Algebraic Manipulation
Once you've set up the initial equation for inversion, your next move is to perform some algebra tricks!

We swapped variables, giving us \( x = -\frac{3}{y} \). At this stage, algebraic manipulation is like unlocking a box to free "y." The goal here is to make "y" the subject of our formula.
  • To do so, multiply both sides by "y" to eliminate the denominator: \( xy = -3 \).
  • Divide by "x" to isolate "y": \( y = -\frac{3}{x} \).
This shows how algebraic manipulation helps transform the equation flawlessly into its new role as an inverse function. Each step requires caution and precision, ensuring we simplify while maintaining balance.
Self-Inverse Functions
Sometimes, after all our work, we find something fascinating: a function that's its own inverse!

When we look at \( f(x) = -\frac{3}{x} \) and its inverse, it turns out they're the same. The function we derived, \( y = -\frac{3}{x} \), loops right back to the original function. This is a perfect example of a self-inverse function.
  • What this indicates is that applying the function once and then again returns us to where we started.
  • Self-inverse functions are rare and symmetrical. Just like a perfect reflection in the mirror!
Recognizing a self-inverse function shows us the true elegance of mathematical symmetry and the fascinating dance of numbers.

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