Find (f°g)⁻¹(x) Easily: Step-by-Step Solution

Hey guys! Today, we're diving deep into the fascinating world of composite functions and their inverses. We've got a specific problem to tackle: Given f(x) = 4x² + x + 2 and g(x) = 5x + 10, we need to figure out (f°g)⁻¹(x). Buckle up, because we're about to break this down step-by-step, making sure you not only get the answer but also understand the why behind it.

Understanding Composite Functions

Before we jump into the inverse, let's quickly recap what a composite function actually is. Think of it like a machine where you feed in an input, and it goes through multiple stages of processing. In mathematical terms, a composite function, written as (f°g)(x), means we're first applying the function g to x, and then we're taking the result and plugging it into the function f. So, (f°g)(x) is the same as f(g(x)). This is crucial to grasp because it dictates the order of operations. The function on the right (g(x) in this case) is applied first, and then the function on the left (f(x)) is applied to the result. It's like a mathematical assembly line, each function doing its part to transform the initial input.

To illustrate further, let’s consider some examples. If we have f(x) = x + 2 and g(x) = 3x, then (f°g)(x) would be f(3x) = (3x) + 2 = 3x + 2. We replaced the 'x' in f(x) with the entire function g(x). Similarly, (g°f)(x) would be g(x + 2) = 3(x + 2) = 3x + 6. Notice that the order matters! (f°g)(x) is not necessarily the same as (g°f)(x). This non-commutative property is a key characteristic of composite functions. Understanding this difference is vital for solving problems involving composite functions and their inverses. For instance, in our problem, we need to find the inverse of (f°g)(x), so we must first determine the expression for (f°g)(x) before we can proceed to find its inverse. This involves substituting g(x) into f(x) and simplifying the resulting expression. Only then can we apply the techniques for finding inverse functions.

Finding (f°g)(x)

Now that we've refreshed our understanding of composite functions, let's find (f°g)(x) for our given functions. Remember, f(x) = 4x² + x + 2 and g(x) = 5x + 10. So, to find (f°g)(x), we need to substitute g(x) into f(x). This means every time we see an 'x' in f(x), we're going to replace it with the entire expression for g(x), which is 5x + 10. Let's get started! We have f(g(x)) = 4(5x + 10)² + (5x + 10) + 2. See how we've replaced each 'x' in f(x) with '(5x + 10)'? This is the heart of finding the composite function. The next step is to simplify this expression, which involves expanding the squared term and combining like terms.

First, we need to expand (5x + 10)². Remember that (a + b)² = a² + 2ab + b². Applying this, we get (5x + 10)² = (5x)² + 2(5x)(10) + 10² = 25x² + 100x + 100. Now we can substitute this back into our expression for f(g(x)): f(g(x)) = 4(25x² + 100x + 100) + (5x + 10) + 2. Next, we distribute the 4: f(g(x)) = 100x² + 400x + 400 + 5x + 10 + 2. Finally, we combine like terms: f(g(x)) = 100x² + (400x + 5x) + (400 + 10 + 2) = 100x² + 405x + 412. There we have it! (f°g)(x) = 100x² + 405x + 412. This quadratic expression represents the composite function. It's a bit more complex than the original functions, but we've broken it down systematically. Now that we have (f°g)(x), we're one step closer to finding its inverse. The next challenge is to determine the inverse function, which involves switching the roles of x and y and solving for y. This process can be more intricate for quadratic functions, so we'll need to be careful with our algebraic manipulations. However, with a clear understanding of the steps involved, we can successfully find the inverse of this composite function.

Unveiling Inverse Functions: The Basics

Before we dive into finding the inverse of our specific composite function, let's take a moment to understand what an inverse function really is. Think of a function as a machine that takes an input (x) and produces an output (y). The inverse function, denoted as f⁻¹(x), is like a machine that does the opposite – it takes the output (y) and gives you back the original input (x). It's like reversing the process. Mathematically, if f(a) = b, then f⁻¹(b) = a. This reciprocal relationship is the essence of inverse functions. The inverse function essentially "undoes" what the original function did. This concept is fundamental to understanding how inverse functions work and how to find them.

Graphically, this "undoing" is represented by a reflection across the line y = x. If you were to plot the graph of a function and its inverse on the same axes, you'd see that they are mirror images of each other with respect to this diagonal line. This visual representation helps to solidify the concept of inverse functions as reflections or reversals of the original function. To find the inverse function algebraically, the general approach involves two main steps: first, replace f(x) with y; second, swap x and y and solve for y. This process effectively reverses the roles of input and output, leading us to the inverse function. However, it's important to note that not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each input corresponds to a unique output, and each output corresponds to a unique input. This condition ensures that the reversal process is well-defined and leads to a valid inverse function.

The Art of Finding the Inverse: A Step-by-Step Guide

Alright, now that we've got the basics down, let's tackle the main challenge: finding (f°g)⁻¹(x). We already know that (f°g)(x) = 100x² + 405x + 412. The process of finding the inverse involves a few key steps, which we'll break down one by one to make it super clear. It's like following a recipe – each step is important, and if we follow them carefully, we'll get the right result!

Step 1: Replace (f°g)(x) with y. This is a simple substitution to make our equation easier to work with. So, we rewrite our composite function as: y = 100x² + 405x + 412. This step transforms the function notation into a more familiar algebraic equation, making it easier to manipulate. Step 2: Swap x and y. This is the heart of finding the inverse – we're reversing the roles of input and output. So, wherever we see a 'y', we replace it with 'x', and wherever we see an 'x', we replace it with 'y'. Our equation now becomes: x = 100y² + 405y + 412. This step effectively represents the inverse relationship, where the original output (x) is now the input, and the original input (y) is now the output. Step 3: Solve for y. This is where things can get a little tricky, especially since we have a quadratic equation. We need to isolate 'y' on one side of the equation. In this case, we have a quadratic equation in the form of ay² + by + c = x. To solve for y, we'll need to use the quadratic formula or complete the square. The quadratic formula is a general solution for quadratic equations, while completing the square involves manipulating the equation to form a perfect square trinomial, which can then be easily solved. We'll explore both methods to find the solution for y. Why is this step the trickiest? Because solving for y often involves algebraic manipulations that can be prone to errors. We need to be careful with signs, factoring, and applying the quadratic formula correctly. For a quadratic equation, there might be two solutions for y, which means the inverse might not be a function in the traditional sense (it might be a relation instead). We'll need to consider the domain and range of the original function to determine which solution, if any, is the correct inverse function. Step 4: Express the result as (f°g)⁻¹(x). Once we've solved for y, we replace 'y' with (f°g)⁻¹(x) to denote the inverse function. This final step puts our solution back into function notation, clearly indicating that we've found the inverse of the composite function. It also serves as a reminder that we've successfully reversed the roles of input and output and found the function that