Verifying Inverse Functions With Composition F(x)=(3/10)x And F⁻¹(x)=(10/3)x

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Hey guys! Today, we're diving deep into the fascinating world of inverse functions and how we can use function composition to verify if we've found the correct inverse. We'll be focusing on a specific example: the function f(x) = (3/10)x and its proposed inverse f⁻¹(x) = (10/3)x. So, buckle up and let's get started!

Understanding Inverse Functions

Before we jump into the nitty-gritty of function composition, let's quickly recap what inverse functions are all about. In essence, an inverse function undoes what the original function does. Think of it like this: if you put a number into a function and get an output, the inverse function will take that output and give you back your original number. This fundamental property is key to understanding how we verify inverses using composition.

Mathematically, we represent the inverse of a function f(x) as f⁻¹(x). A crucial requirement for a function to have an inverse is that it must be one-to-one, meaning that each input corresponds to a unique output, and vice-versa. Graphically, this translates to the function passing the horizontal line test. Now, how do we prove that a given function is indeed the inverse of another? That's where function composition comes in!

Function Composition: The Key to Verification

Function composition is the process of applying one function to the result of another. We denote the composition of f with g as (f ∘ g)(x), which means we first apply g(x) and then apply f to the result. In our context of inverse functions, the magic happens when we compose a function with its inverse (in both orders!).

The core principle is this: If f⁻¹(x) is truly the inverse of f(x), then composing them in either order should result in the identity function, which is simply x. In other words:

  • (f⁻¹ ∘ f)(x) = x
  • (f ∘ f⁻¹)(x) = x

If both of these conditions hold true, then we can confidently declare that f⁻¹(x) is indeed the inverse of f(x). This is the litmus test for inverse functions! Let's put this into practice with our example.

Verifying the Inverse: f(x) = (3/10)x and f⁻¹(x) = (10/3)x

Alright, let's get our hands dirty and verify if f⁻¹(x) = (10/3)x is the inverse of f(x) = (3/10)x. We'll do this by performing the two compositions we discussed earlier.

Step 1: Finding (f⁻¹ ∘ f)(x)

Remember, (f⁻¹ ∘ f)(x) means we first apply f(x) and then apply f⁻¹(x) to the result. Let's break it down:

  1. Start with f(x) = (3/10)x.
  2. Now, we need to plug this entire expression into f⁻¹(x). So, we replace the x in f⁻¹(x) = (10/3)x with (3/10)x.
  3. This gives us f⁻¹(f(x)) = f⁻¹((3/10)x) = (10/3) * (3/10)x.
  4. Now, let's simplify! Notice that the (10/3) and (3/10) cancel each other out, leaving us with x.
  5. Therefore, (f⁻¹ ∘ f)(x) = x. Hooray! One condition down.

Step 2: Finding (f ∘ f⁻¹)(x)

Now, let's tackle the other composition, (f ∘ f⁻¹)(x). This time, we first apply f⁻¹(x) and then apply f(x) to the result.

  1. Start with f⁻¹(x) = (10/3)x.
  2. We need to plug this expression into f(x). So, we replace the x in f(x) = (3/10)x with (10/3)x.
  3. This gives us f(f⁻¹(x)) = f((10/3)x) = (3/10) * (10/3)x.
  4. Again, we simplify! The (3/10) and (10/3) cancel each other out, leaving us with x.
  5. Therefore, (f ∘ f⁻¹)(x) = x. Double hooray! Both conditions are met.

Conclusion: The Inverse is Verified!

We've successfully shown that both (f⁻¹ ∘ f)(x) = x and (f ∘ f⁻¹)(x) = x. This confirms, without a doubt, that f⁻¹(x) = (10/3)x is indeed the inverse of f(x) = (3/10)x. Function composition is a powerful tool, guys, and this example perfectly illustrates how we can use it to verify inverse functions. Understanding this concept is crucial for mastering more advanced mathematical topics. Keep practicing, and you'll become inverse function pros in no time! Remember the key takeaway: composing a function with its true inverse will always result in the identity function, x. This is the golden rule of inverse function verification! So, the next time you're faced with verifying an inverse, remember this process and you'll be golden. Now go forth and conquer those functions!

Practice Problems

To solidify your understanding, try verifying the inverses of the following functions using composition:

  1. f(x) = 2x + 1, f⁻¹(x) = (x - 1)/2
  2. g(x) = x³, g⁻¹(x) = ∛x
  3. h(x) = (x + 5)/3, h⁻¹(x) = 3x - 5

Work through these problems step-by-step, and you'll become even more confident in your ability to work with inverse functions.

Further Exploration

If you're interested in delving deeper into the world of inverse functions, consider exploring these topics:

  • Finding Inverses Algebraically: Learn the steps involved in finding the inverse of a function when given its equation.
  • Graphing Inverse Functions: Discover the relationship between the graphs of a function and its inverse.
  • Inverse Trigonometric Functions: Explore the inverses of trigonometric functions, such as sine, cosine, and tangent.

By expanding your knowledge in these areas, you'll gain a more comprehensive understanding of inverse functions and their applications.

Key Takeaways

  • An inverse function undoes what the original function does.
  • To verify if f⁻¹(x) is the inverse of f(x), check if (f⁻¹ ∘ f)(x) = x and (f ∘ f⁻¹)(x) = x.
  • If both compositions result in x, then f⁻¹(x) is indeed the inverse of f(x).
  • Function composition is a powerful tool for verifying inverse functions.

Keep these key takeaways in mind, and you'll be well on your way to mastering inverse functions!