Function Composition: Finding (f ○ G)(x) And (g ○ F)(x)
Hey guys! Let's dive into the fascinating world of function composition. We've got two functions here, f(x) = 4x + 1 and g(x) = x² - 2x + 3, and we're going to explore what happens when we combine them. Function composition is like a mathematical assembly line – we feed the output of one function into another. So, buckle up, and let's get started!
Understanding Function Composition
Before we jump into the calculations, let's make sure we're all on the same page about what function composition actually means. When we see something like (f ○ g)(x), it means we're plugging the entire function g(x) into the function f(x). Think of it as f(g(x)). Similarly, (g ○ f)(x) means we're plugging f(x) into g(x), or g(f(x)). The order matters a lot here, guys! It's like putting on your socks before your shoes – you gotta do it in the right order to get the desired result.
So, why is this important? Function composition allows us to create more complex functions from simpler ones. It's a fundamental concept in mathematics and has applications in various fields, including computer science, engineering, and physics. Imagine you have a function that converts temperature from Celsius to Fahrenheit, and another function that converts Fahrenheit to Kelvin. By composing these functions, you can directly convert Celsius to Kelvin! Pretty cool, right?
Now, let's talk about the domain and range of composite functions. The domain of (f ○ g)(x) is the set of all x in the domain of g such that g(x) is in the domain of f. In simpler terms, we need to make sure that the output of g is a valid input for f. Similarly, the range of the composite function depends on the ranges of both f and g. Understanding these concepts is crucial for working with function composition effectively. We need to consider the restrictions each function imposes on the other.
Calculating (f ○ g)(x)
Okay, let's get our hands dirty and calculate (f ○ g)(x). Remember, this means we're going to substitute g(x) into f(x). Our functions are:
- f(x) = 4x + 1
- g(x) = x² - 2x + 3
So, to find (f ○ g)(x), we replace the 'x' in f(x) with the entire expression for g(x). This gives us:
(f ○ g)(x) = f(g(x)) = 4(g(x)) + 1 = 4(x² - 2x + 3) + 1
Now, we just need to simplify this expression by distributing the 4 and combining like terms:
(f ○ g)(x) = 4x² - 8x + 12 + 1 = 4x² - 8x + 13
So, (f ○ g)(x) = 4x² - 8x + 13. Looking at the given options, statement A, which says (f ○ g)(x) = 4x² - 8x + 13, is indeed correct! We've successfully composed the functions and arrived at the correct expression. High five!
Calculating (g ○ f)(x)
Now, let's flip the script and find (g ○ f)(x). This time, we're substituting f(x) into g(x). Remember, the order is super important! We're not going to get the same result as (f ○ g)(x), so pay close attention. We have:
- f(x) = 4x + 1
- g(x) = x² - 2x + 3
To find (g ○ f)(x), we replace the 'x' in g(x) with the expression for f(x):
(g ○ f)(x) = g(f(x)) = (f(x))² - 2(f(x)) + 3 = (4x + 1)² - 2(4x + 1) + 3
This looks a little more complicated, but don't worry, we'll break it down step by step. First, we need to expand the squared term:
(4x + 1)² = (4x + 1)(4x + 1) = 16x² + 8x + 1
Now, let's substitute this back into our expression and distribute the -2:
(g ○ f)(x) = 16x² + 8x + 1 - 8x - 2 + 3
Finally, we combine like terms:
(g ○ f)(x) = 16x² + 2
Wait a minute! Looking at the given options, statement B says (g ○ f)(x) = 16x² + 8x - 2, but we got (g ○ f)(x) = 16x² + 2. So, statement B is incorrect. It's crucial to double-check our work, guys, as even a small mistake can lead to a wrong answer!
Evaluating (g ○ f)(1)
Okay, let's move on to the next part of the problem. We need to evaluate (g ○ f)(1). This means we're going to plug in x = 1 into the expression we found for (g ○ f)(x). We already know that (g ○ f)(x) = 16x² + 2, so let's substitute x = 1:
(g ○ f)(1) = 16(1)² + 2 = 16(1) + 2 = 16 + 2 = 18
So, (g ○ f)(1) = 18. Statement C says (g ○ f)(1) = 18, which matches our result. So, statement C is correct! We're on a roll, guys!
Analyzing (g ○ f)(x) = ...
Finally, let's address the last part of the problem, which asks about the expression for (g ○ f)(x). We've already done the hard work here! We calculated (g ○ f)(x) earlier and found it to be 16x² + 2. This highlights the importance of carefully calculating and simplifying expressions when dealing with function composition. A solid understanding of the process is key to getting the right answers.
Key Takeaways
Let's recap what we've learned in this problem:
- Function composition involves plugging one function into another.
- The order of composition matters – (f ○ g)(x) is generally not the same as (g ○ f)(x).
- To find (f ○ g)(x), substitute g(x) into f(x).
- To find (g ○ f)(x), substitute f(x) into g(x).
- Carefully expand and simplify expressions to avoid errors.
- Evaluating composite functions at a specific value involves substituting the value into the composite function's expression.
Understanding function composition is a crucial stepping stone to more advanced mathematical concepts. It's a powerful tool for building complex models and solving real-world problems. So, keep practicing, guys, and you'll become function composition masters in no time!
Conclusion
In this problem, we explored the concept of function composition, calculated (f ○ g)(x) and (g ○ f)(x), evaluated (g ○ f)(1), and identified the correct statements. We found that statements A and C are true, while statement B is false. Remember, guys, the key to success in function composition is a clear understanding of the process, careful calculations, and a healthy dose of practice. Keep up the great work, and I'll see you in the next mathematical adventure! Remember to always double-check your work and break down complex problems into smaller, more manageable steps. You've got this!
