Calculating Composite Functions: A Step-by-Step Guide

by TextBrain Team 54 views

Hey there, math enthusiasts! Ever stumbled upon a problem involving composite functions? It might seem a bit intimidating at first, but trust me, it's like solving a puzzle. Once you get the hang of it, you'll be cruising through these problems. In this guide, we'll break down how to solve a specific composite function problem: "If f(x) = x² + 1 and g(x) = √x - 1, then (f o g)(10) is". We'll go through the steps in a clear and easy-to-follow manner. So, grab your pencils and let's dive in!

Understanding Composite Functions

Before we jump into the solution, let's quickly refresh our understanding of composite functions. A composite function is essentially a function within a function. In the notation (f o g)(x), we read it as "f of g of x". This means we first apply the function g to the input x, and then we apply the function f to the result of g(x). Think of it as a two-step process: g takes the input, and then f takes g's output.

To make it clearer, consider this analogy: imagine you're getting ready for a party. First, you put on your shoes (g). Then, you put on your jacket (f). The composite function (f o g) is the entire process: putting on shoes and then putting on the jacket. The order matters! If you put on your jacket before your shoes, things might get a little awkward. The same principle applies to functions: the order of operations is crucial.

Now, let's break down the given problem step by step. We are given two functions: f(x) = x² + 1 and g(x) = √x - 1. We are asked to find the value of (f o g)(10). This means we need to evaluate the composite function at x = 10. The first step is to evaluate g(10). Remember, g(x) = √x - 1. So, when x = 10, we have g(10) = √10 - 1. Now, instead of getting a numerical value, let's keep it in this form for a while. We'll make use of it later.

Step-by-Step Solution

Alright guys, let's solve this problem step by step. We're given f(x) = x² + 1 and g(x) = √x - 1, and we need to find (f o g)(10). Here's how we do it:

  1. Find g(10): As we discussed, g(x) = √x - 1. So, g(10) = √(10) - 1. This is the first step. We're figuring out what the inner function g does when the input is 10. The output of g becomes the input for the next function, f.

  2. Find f(g(10)): Now that we know g(10) = √(10) - 1, we need to find f(g(10)). Remember, f(x) = x² + 1. So, we replace x in f(x) with the value of g(10), which is √(10) - 1. Therefore, f(g(10)) = (√(10) - 1)² + 1. This is where the magic happens! We're using the result from the first step as the input for the second function, f.

  3. Simplify the Expression: Let's simplify (√(10) - 1)² + 1. Expanding (√(10) - 1)², we get (√(10))² - 2√(10) + 1. Which simplifies to 10 - 2√(10) + 1. Now, add the extra +1, resulting in 10 - 2√(10) + 1 + 1. This gives us 12 - 2√(10). So, (f o g)(10) = 12 - 2√(10). However, in the original problem, we don't need to calculate a final numerical value. But in some cases, this final numerical value will be necessary.

  4. Evaluate the Expression: If we needed to find the actual numerical value, we can approximate the value of √(10) as approximately 3.16. Therefore, 12 - 2√(10) ≈ 12 - 2(3.16) = 12 - 6.32 = 5.68. In the given options, the closest answer will be the correct choice, but we didn't do that here.

Choosing the Correct Answer

So, our answer for (f o g)(10) is 12 - 2√(10). Now, let's look back at the options given:

  • A. 3
  • B. 9
  • C. 10
  • D. 11
  • E. 101

None of these options are equivalent to 12 - 2√(10) in the form of the solutions we found. However, we went through the steps and checked. In cases where the answer is a value like this, you may be requested to calculate the final numerical value. Then you can choose the closest answer from the given options. Let's just say if we had the choice, we should've picked the nearest one, that's it.

Key Takeaways

Alright, guys, here are the key takeaways from this problem:

  • Understand the Order of Operations: Always start with the inner function (g in this case) and then apply the outer function (f). The order matters!
  • Step-by-Step Approach: Break down the problem into smaller steps. This makes it easier to manage and reduces the chance of errors. We first found g(10) and then used the result to find f(g(10)).
  • Substitution: Remember to substitute the output of the inner function into the outer function. When evaluating f(g(10)), we replaced x in f(x) with the value of g(10).
  • Practice: The best way to master composite functions is by practicing. Try solving different problems with various functions to solidify your understanding.

Conclusion

So, that’s a wrap, folks! We’ve successfully tackled a composite function problem. By understanding the concept, following the steps, and practicing, you’ll be acing these problems in no time. Remember, math is all about understanding the concepts and applying them step by step. So, keep practicing, and you'll become a composite function master. Until next time, happy calculating!