Inverse Function: Find F⁻¹(x), Domain & Range

Hey guys! Today, we're diving into the fascinating world of inverse functions. Specifically, we're going to tackle the problem of finding the inverse of a one-to-one function, determining its domain and range, and expressing these in interval notation. Let's use the function ${ f(x) = \frac{6x - 7}{7x + 4} }$ as our example. So, buckle up, and let's get started!

Understanding One-to-One Functions and Inverses

Before we jump into the nitty-gritty, let's quickly recap what one-to-one functions and their inverses are all about. A one-to-one function, also known as an injective function, is a function where each element of the range corresponds to exactly one element in the domain. In simpler terms, no two different inputs produce the same output. This property is crucial because only one-to-one functions have inverses. An inverse function, denoted as ${ f^{-1}(x) }$, essentially 'undoes' what the original function does. If ${ f(a) = b }$, then ${ f^{-1}(b) = a }$. Think of it like a reverse operation; if you put something in and get something out, the inverse function takes that output and gives you back the original input. This reciprocal relationship is key to understanding how to find and work with inverse functions. Graphically, a function and its inverse are reflections of each other across the line ${ y = x }$. This visual representation can be incredibly helpful in understanding the nature of inverse functions and their properties.

To determine if a function is one-to-one, we can use the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one and does not have an inverse. Alternatively, we can algebraically verify the one-to-one property by showing that if ${ f(x_1) = f(x_2) }$, then ${ x_1 = x_2 }$. This means that if two inputs produce the same output, those inputs must be the same. Understanding these fundamental concepts is essential for successfully navigating the process of finding and analyzing inverse functions. So, now that we have a solid foundation, let's move on to the specific steps involved in finding the inverse of our function ${ f(x) = \frac{6x - 7}{7x + 4} }$.

Step-by-Step Guide to Finding the Inverse Function

Alright, let's dive into the fun part: finding the inverse function ${ f^{-1}(x) }$ for ${ f(x) = \frac{6x - 7}{7x + 4} }$. We'll break it down into simple, easy-to-follow steps.

1. Replace ${ f(x) }$ with ${ y }$: This is a simple substitution to make the equation easier to work with. So, we rewrite the function as: ${ y = \frac{6x - 7}{7x + 4} }$ This step is crucial because it sets up the equation in a form that allows us to easily swap the variables in the next step. It's a simple change, but it makes the subsequent algebraic manipulations much clearer and more manageable.

2. Swap ${ x }$ and ${ y }$: This is the heart of finding the inverse! We're essentially reversing the roles of input and output. The equation now becomes: ${ x = \frac{6y - 7}{7y + 4} }$ By swapping ${ x }$ and ${ y }$, we're reflecting the function across the line ${ y = x }$, which is the geometric representation of finding the inverse. This step is the core concept behind inverse functions, as it directly implements the idea of reversing the function's operation. It's like taking the input-output relationship and flipping it around, making the old output the new input and vice versa.

3. Solve for ${ y }$: This is where the algebra comes in. We need to isolate ${ y }$ on one side of the equation. Here’s how we do it:

   • Multiply both sides by ${ (7y + 4) }$ to get rid of the denominator: ${ x(7y + 4) = 6y - 7 }$
   • Distribute the ${ x }$ on the left side: ${ 7xy + 4x = 6y - 7 }$
   • Rearrange the equation to group the terms containing ${ y }$ on one side and the other terms on the other side: ${ 7xy - 6y = -4x - 7 }$
   • Factor out ${ y }$ from the left side: ${ y(7x - 6) = -4x - 7 }$
   • Finally, divide both sides by ${ (7x - 6) }$ to isolate ${ y }$: ${ y = \frac{-4x - 7}{7x - 6} }$ Each of these algebraic manipulations is a step towards isolating ${ y }$, which represents the inverse function. It's a process of carefully unwinding the equation until we have ${ y }$ expressed in terms of ${ x }$. This requires a solid understanding of algebraic principles and the ability to manipulate equations effectively. The goal is to perform operations that maintain the equality while gradually isolating the variable we're interested in.

4. Replace ${ y }$ with ${ f^{-1}(x) }$: This is the final touch! We replace ${ y }$ with the inverse function notation: ${ f^{-1}(x) = \frac{-4x - 7}{7x - 6} }$ So, we've found our inverse function! This step is important because it formally represents the solution as the inverse function, using the correct notation. It's a way of saying,