Function Composition: Solve F(g(x)) = 5 Step-by-Step

by Felix Dubois 53 views

Hey guys! Let's dive into the fascinating world of function composition with a classic math problem. We're given two functions, f(x) = x² + 1 and g(x) = x - 2, defined on the set of real numbers (R). Our mission? To find f(g(x)) (that's f composed with g, often read as "f of g of x") and then pinpoint the x values that make f(g(x)) equal to 5. Buckle up, because we're about to embark on a mathematical adventure!

Navigating Function Composition: Finding f(g(x))

So, what exactly is function composition? Simply put, it's like a mathematical assembly line. We take the output of one function (our g in this case) and feed it as the input into another function (f). To find f(g(x)), we'll substitute the entire function g(x) into the variable x of function f(x). In other words, wherever we see an x in f(x), we'll replace it with (x - 2). This is a core concept in understanding how functions interact and build upon each other.

Let's break it down step-by-step:

  1. Start with f(x): Our function f(x) is defined as x² + 1. This means that whatever we put into f, it squares it and then adds 1. Think of it like a little machine that takes a number, multiplies it by itself, and then adds 1.
  2. Substitute g(x) into f(x): Now, we replace the x in f(x) with the entire function g(x), which is (x - 2). This gives us f(g(x)) = (x - 2)² + 1. This substitution is the heart of function composition – we're literally plugging one function into another.
  3. Expand and Simplify: To make things clearer, let's expand the square and simplify the expression. (x - 2)² expands to x² - 4x + 4. So, f(g(x)) = x² - 4x + 4 + 1. Combining the constants, we get f(g(x)) = x² - 4x + 5. This is the composed function – it represents the combined effect of first applying g and then applying f. We've now successfully navigated the first part of our challenge! Understanding this process is crucial for tackling more complex problems in calculus and other advanced math topics.

Cracking the Code: Solving f(g(x)) = 5

Now that we've found f(g(x)) = x² - 4x + 5, the next challenge is to find the values of x that make this expression equal to 5. This is like reverse-engineering the process – we know the final output (5) and we need to find the input(s) (x) that produce it. It’s a classic algebraic puzzle that puts our problem-solving skills to the test.

Here's how we'll tackle it:

  1. Set up the Equation: We start by setting our expression for f(g(x)) equal to 5: x² - 4x + 5 = 5. This is the equation we need to solve to find the x values that satisfy the condition.
  2. Simplify the Equation: The first step in solving any equation is to simplify it. Subtracting 5 from both sides of the equation gives us x² - 4x = 0. This simplified form is much easier to work with.
  3. Factor the Equation: Now, we factor out a common factor of x from the left side: x(x - 4) = 0. Factoring is a powerful technique for solving equations, as it allows us to break down a complex expression into simpler parts.
  4. Solve for x: For the product of two factors to be zero, at least one of them must be zero. So, either x = 0 or (x - 4) = 0. Solving the second equation gives us x = 4. These are the two solutions to our problem!
  5. Check your solution: You can plug each value back into the original equation to see if it is valid.

Therefore, the values of x that satisfy f(g(x)) = 5 are x = 0 and x = 4. We've successfully cracked the code and found the solutions! This process demonstrates the importance of algebraic manipulation and problem-solving techniques in mathematics.

Key Takeaways and Real-World Connections

This example highlights some fundamental concepts in mathematics:

  • Function Composition: We've seen how to combine functions by plugging one into another. This is a crucial concept in calculus and other advanced math topics. Understanding function composition allows us to model complex relationships and systems.
  • Solving Equations: We've practiced solving quadratic equations by factoring. This is a core skill in algebra and is essential for many applications in science and engineering. The ability to solve equations is a fundamental tool for problem-solving in many fields.
  • Mathematical Thinking: We've broken down a problem into smaller, manageable steps. This is a valuable skill in any field. Mathematical thinking emphasizes logical reasoning, problem decomposition, and systematic approaches to finding solutions. It’s a skill that transcends mathematics and applies to a wide range of challenges.

Function composition might seem abstract, but it has real-world connections. For example, think about a manufacturing process. One function might represent the cost of raw materials, and another might represent the cost of labor. Composing these functions could give you the total cost of production. In computer programming, functions are often composed to build complex algorithms and software. Understanding these mathematical concepts provides a powerful framework for analyzing and solving problems in various domains.

By understanding function composition and equation-solving techniques, we can tackle a wide range of mathematical problems and real-world scenarios. So, keep practicing, keep exploring, and keep those mathematical gears turning!

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