Inverse Functions & Equations: A Step-by-Step Guide

by Rajiv Sharma 52 views

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of functions. We'll be tackling a problem that involves finding the inverse of a function and then using it to solve an equation. Don't worry, it's not as scary as it sounds! We'll break it down step by step, so you can follow along easily. Let's get started!

The Problem at Hand

We've got two functions to play with:

  • f(x) = x/2 - 2, which is a linear function.
  • g(x) = x^2 - 4x, which is a quadratic function.

Our mission, should we choose to accept it (and we do!), is twofold:

a. Find the inverse of the function f(x), which we'll denote as f⁻¹(x). This is like finding the "undo" button for the function f(x).

b. Solve the equation f(g(x)) = f⁻¹(x). This means we need to plug g(x) into f(x), set the result equal to the inverse function we found in part (a), and then solve for x. And, importantly, we need to leave our answers as exact values – no rounding to decimals here!

Part a: Finding the Inverse Function, f⁻¹(x)

Okay, let's get our hands dirty and find that inverse function. Remember, the inverse function essentially reverses the operation of the original function. Here's how we do it:

  1. Replace f(x) with y: This makes the algebra a bit cleaner. So, we rewrite f(x) = x/2 - 2 as y = x/2 - 2.
  2. Swap x and y: This is the key step in finding the inverse. We're essentially reversing the roles of input and output. Our equation now becomes x = y/2 - 2.
  3. Solve for y: Now, we need to isolate y on one side of the equation. Let's add 2 to both sides: x + 2 = y/2. Then, multiply both sides by 2: 2(x + 2) = y. Finally, distribute the 2: 2x + 4 = y.
  4. Replace y with f⁻¹(x): This is just a notational step to show that we've found the inverse function. So, we have f⁻¹(x) = 2x + 4.

Ta-da! We've found the inverse function. f⁻¹(x) = 2x + 4. This means that if we plug a value into f(x) and then plug the result into f⁻¹(x), we should get back our original value (and vice versa). Think of it like putting on your shoes and then taking them off – you end up back where you started.

Part b: Solving the Equation f(g(x)) = f⁻¹(x)

Alright, now for the main event: solving the equation f(g(x)) = f⁻¹(x). This is where things get a little more interesting, but don't worry, we'll take it one step at a time.

  1. Find f(g(x)): This means we need to plug the function g(x) into the function f(x). Remember, g(x) = x² - 4x and f(x) = x/2 - 2. So, we replace the x in f(x) with g(x): f(g(x)) = (x² - 4x) / 2 - 2

    Let's simplify this a bit by distributing the division: f(g(x)) = x²/2 - 2x - 2

  2. Set up the equation: Now we have f(g(x)) = x²/2 - 2x - 2 and we know f⁻¹(x) = 2x + 4. We can set these equal to each other: x²/2 - 2x - 2 = 2x + 4

  3. Solve the equation: This looks like a quadratic equation, so let's get everything on one side and set it equal to zero. First, subtract 2x and 4 from both sides: x²/2 - 4x - 6 = 0

    To get rid of the fraction, let's multiply the entire equation by 2: x² - 8x - 12 = 0

    Now we have a standard quadratic equation. We can try to factor it, but it doesn't seem to factor nicely. So, we'll use the quadratic formula. Remember the quadratic formula? For an equation of the form ax² + bx + c = 0, the solutions are given by:

    x = (-b ± √(b² - 4ac)) / 2a

    In our case, a = 1, b = -8, and c = -12. Let's plug these values into the quadratic formula:

    x = (8 ± √((-8)² - 4 * 1 * -12)) / (2 * 1) x = (8 ± √(64 + 48)) / 2 x = (8 ± √112) / 2

    We can simplify the square root a bit. 112 can be factored as 16 * 7, and the square root of 16 is 4: x = (8 ± 4√7) / 2

    Finally, we can divide both terms in the numerator by 2: x = 4 ± 2√7

Boom! We've got our solutions. The solutions to the equation f(g(x)) = f⁻¹(x) are x = 4 + 2√7 and x = 4 - 2√7. These are exact values, just like the problem asked for. Excellent work, guys!

Wrapping Up

So, there you have it! We successfully found the inverse of a function and solved a tricky equation involving function composition and inverses. Remember, the key to these types of problems is to break them down into smaller, manageable steps. Don't be intimidated by the complexity; just take it one step at a time. You've got this!

Understanding functions, their inverses, and how to manipulate them is a crucial skill in mathematics. It pops up in all sorts of areas, from calculus to cryptography. So, pat yourselves on the back for tackling this problem head-on. You're one step closer to mastering the mathematical universe!

If you enjoyed this breakdown and want to see more math explorations, let me know in the comments below! What other topics are you curious about? What challenges are you facing? I'm here to help you on your mathematical journey. Keep exploring, keep learning, and keep having fun with math!

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