Solving For X In Composite And Inverse Functions If F(x)=3x+5 And F . G(x+2)=12x+17

In the fascinating realm of mathematics, functions serve as fundamental building blocks, mapping inputs to outputs and weaving intricate relationships between variables. Among the diverse types of functions, composite functions and inverse functions hold a special allure, often presenting intriguing puzzles that challenge our analytical prowess. This article delves into a captivating problem involving these very concepts, guiding you through a step-by-step solution while illuminating the underlying principles at play.

Deconstructing Composite Functions: A Journey into Function Composition

At the heart of our mathematical expedition lies the concept of composite functions. In essence, a composite function is a function that operates on the output of another function. Imagine two functions, $f(x)$ and $g(x)$. The composite function, denoted as $f(g(x))$, represents the application of the function $f$ to the result of applying the function $g$ to the input $x$. This composition creates a chain reaction, where the output of one function becomes the input of the next.

In the given problem, we are presented with two functions: $f(x) = 3x + 5$ and a composite function $f(g(x+2)) = 12x + 17$. Our mission is to unravel the intricate relationship between these functions and ultimately determine the value of $x$ that satisfies the condition $g^{-1}(x) = 88$. This journey requires us to dissect the composite function, isolate the inner function $g(x)$, and then venture into the realm of inverse functions.

The beauty of composite functions lies in their ability to model complex processes by breaking them down into simpler, sequential steps. For instance, consider the process of baking a cake. We can represent the individual steps, such as mixing ingredients, baking, and frosting, as functions. The composite function would then represent the entire cake-baking process, where the output of one step becomes the input of the next. This modularity allows us to analyze and optimize each step individually, leading to a more efficient and delectable outcome. In our mathematical problem, understanding the composition of $f$ and $g$ is crucial to unlocking the solution. We must carefully consider how the output of $g(x+2)$ becomes the input of $f$, and how this interaction shapes the final result. By meticulously dissecting this composite function, we can pave the way for unraveling the value of $x$ that satisfies the given condition.

Inverse Functions: Reversing the Mapping

To fully appreciate the problem at hand, we must also grasp the concept of inverse functions. An inverse function, denoted as $f^{-1}(x)$, is a function that "undoes" the action of the original function $f(x)$. In other words, if $f(a) = b$, then $f^{-1}(b) = a$. The inverse function essentially reverses the mapping performed by the original function, taking the output back to its original input.

Consider the function $f(x) = 2x$. This function doubles the input value. Its inverse function, $f^{-1}(x) = x/2$, halves the input value, effectively undoing the doubling operation. Inverse functions are invaluable tools in mathematics, allowing us to solve equations, analyze relationships, and gain deeper insights into the behavior of functions. In our problem, the condition $g^{-1}(x) = 88$ involves the inverse function of $g(x)$. This means that we need to find the value of $x$ that, when inputted into $g^{-1}$, produces an output of 88. To achieve this, we must first determine the explicit form of the function $g(x)$, which requires us to carefully dissect the composite function $f(g(x+2))$. The interplay between composite functions and inverse functions is a recurring theme in mathematics, and mastering these concepts is crucial for tackling a wide range of problems. By understanding how functions compose and how they can be reversed, we equip ourselves with powerful tools for mathematical exploration.

Step-by-Step Solution: Unveiling the Value of x

Now, let's embark on the journey of solving the problem. Our objective is to find the value of $x$ that satisfies the condition $g^{-1}(x) = 88$, given the functions $f(x) = 3x + 5$ and $f(g(x+2)) = 12x + 17$.

1. Dissecting the Composite Function: Our first step is to unravel the composite function $f(g(x+2))$. We know that $f(x) = 3x + 5$. Therefore, $f(g(x+2))$ means we substitute $g(x+2)$ into the expression for $f(x)$. This gives us:

   $$f(g(x+2)) = 3[g(x+2)] + 5$$

   We are also given that $f(g(x+2)) = 12x + 17$. Equating these two expressions, we get:

   $$3[g(x+2)] + 5 = 12x + 17$$

2. Isolating g(x+2): Now, we need to isolate the function $g(x+2)$. To do this, we subtract 5 from both sides of the equation:

   $$3[g(x+2)] = 12x + 12$$

   Next, we divide both sides by 3:

   $$g(x+2) = 4x + 4$$

3. Finding g(x): To find $g(x)$, we perform a substitution. Let $y = x + 2$. Then, $x = y - 2$. Substituting this into the expression for $g(x+2)$, we get:

   $$g(y) = 4(y - 2) + 4$$

   Simplifying, we obtain:

   $$g(y) = 4y - 8 + 4$$

   $$g(y) = 4y - 4$$

   Replacing $y$ with $x$, we have:

   $$g(x) = 4x - 4$$

4. Finding the Inverse Function g⁻¹(x): To find the inverse function $g^{-1}(x)$, we first replace $g(x)$ with $y$:

   $$y = 4x - 4$$

   Next, we swap $x$ and $y$:

   $$x = 4y - 4$$

   Now, we solve for $y$:

   $$x + 4 = 4y$$

   y = rac{x + 4}{4}

   Therefore, the inverse function is:

   g^{-1}(x) = rac{x + 4}{4}

5. Solving for x: We are given that $g^{-1}(x) = 88$. Substituting the expression for $g^{-1}(x)$, we get:

   rac{x + 4}{4} = 88

   Multiplying both sides by 4, we have:

   $$x + 4 = 352$$

   Subtracting 4 from both sides, we obtain:

   $$x = 348$$

Therefore, the value of $x$ that satisfies the condition $g^{-1}(x) = 88$ is 348.

Conclusion: The Interplay of Functions

In this mathematical exploration, we successfully navigated the realms of composite functions and inverse functions to uncover the value of $x$ that satisfies a given condition. By meticulously dissecting the composite function, isolating the inner function, and venturing into the world of inverse functions, we demonstrated the power of analytical thinking and problem-solving in mathematics. The interplay between composite functions and inverse functions is a testament to the interconnectedness of mathematical concepts, highlighting the importance of a holistic understanding of these fundamental building blocks. As we continue our mathematical journey, may we embrace the challenges and unravel the mysteries that lie ahead, armed with the tools and insights gained from this exploration.

Understanding the Problem

The question presents a scenario involving composite functions and inverse functions. We are given two functions: $f(x) = 3x + 5$ and a composite function $f(g(x+2)) = 12x + 17$. The objective is to find the value of $x$ for which the inverse of function $g$, denoted as $g^{-1}(x)$, equals 88. To solve this, we need to first determine the function $g(x)$, then find its inverse $g^{-1}(x)$, and finally solve the equation $g^{-1}(x) = 88$ for $x$.

This problem requires a solid understanding of function composition and inverse functions. Function composition involves applying one function to the result of another. In this case, $f(g(x+2))$ means we first evaluate $g(x+2)$, and then use that result as the input for the function $f$. An inverse function, on the other hand,