Composition Function $(g \circ F)(1)$: Step-by-Step Calculation

Hey math enthusiasts! Today, we're diving into the world of composition functions. Specifically, we're going to figure out the value of $(g \circ f)(1)$ when we're given the functions $f(x) = 8x - 1$ and $g(x) = 2x^2 + 3$. Don't worry, it sounds a lot more complicated than it actually is. We'll break it down into easy-to-follow steps, and by the end, you'll be a pro at this! So, buckle up, grab your calculators (or your brains!), and let's get started. Understanding composition functions is a fundamental concept in algebra, and it's super useful in a bunch of different areas of math and science. So, let's start with some basic knowledge.

Understanding Composition Functions

Alright guys, before we jump into the actual problem, let's make sure we're all on the same page about what a composition function actually is. In simple terms, a composition function is a function that applies one function to the result of another function. Think of it like a chain reaction: you first apply one function, and then you take the output of that function and plug it into another function. The notation $(g \circ f)(x)$ means that you first apply the function $f$ to $x$, and then you apply the function $g$ to the result of $f(x)$. It's like a mathematical assembly line! So if we see $(g \circ f)(1)$, it means we first find the value of $f(1)$, and then we use that value as the input for the function $g$. Easy peasy, right? Now, let's see this in action with our given functions. Understanding this is crucial for success, so make sure you're following along. Composition of functions is a really powerful idea that can be used in many fields.

Step-by-Step Solution

Now that we know the basics of composition functions, let's solve the problem. We're given $f(x) = 8x - 1$ and $g(x) = 2x^2 + 3$, and we want to find $(g \circ f)(1)$. Here’s how we do it step by step:

Step 1: Find $f(1)$

First, we need to figure out what $f(1)$ is. This means we substitute $x = 1$ into the function $f(x)$. So, we have:

$f(1) = 8(1) - 1$

$f(1) = 8 - 1$

$f(1) = 7$

See? That wasn't so bad, was it? We now know that $f(1) = 7$. This is a crucial step because this value becomes the input for our next function, $g$. Remember that the goal here is to be able to solve problems and understand them. We have to take things step by step. Always go back and make sure you understand what you have done before you proceed.

Step 2: Find $g(f(1))$

Next, we need to find $g(f(1))$, which is the same as $g(7)$ because we know that $f(1) = 7$. Now, we substitute $x = 7$ into the function $g(x)$.

$g(7) = 2(7)^2 + 3$

$g(7) = 2(49) + 3$

$g(7) = 98 + 3$

$g(7) = 101$

And there you have it! We've found that $g(f(1)) = 101$. We used the value of $f(1)$ and substituted it in the function $g$. That is how we found the final answer. The core of understanding composition functions lies in knowing how to apply the functions in the correct order and correctly substitute the values. Congratulations, you have the solution!

The Answer

So, the final answer to $(g \circ f)(1)$ is $101$. We went through the steps, found $f(1)$, and then used that output as the input for $g$. That is how we tackled our initial problem. Great job, everyone! Now, let's relate that answer to the choices.

• The provided choices are:

  • 8
  • 9
  • 10
  • 11
  • 12

• However, our calculated answer is 101.

  • Therefore, none of the given options match our correct answer. It looks like there may be an error in the answer choices.

Conclusion and Further Exploration

So, there you have it, guys! We've successfully found the value of $(g \circ f)(1)$ by breaking it down into simple steps. This is how you understand composition functions. We started with our functions, found $f(1)$, and then plugged that result into $g$. Always remember that the key is to work step by step and understand what you are doing. Composition functions might seem tricky at first, but with practice, they become a piece of cake. Now that you've got the basics down, try practicing with different functions and values. You can also explore more complex composition functions, where you might have to compose more than two functions together. There are many online resources and textbooks with a lot of extra practice problems! You'll find a lot of interesting problems that will challenge you and improve your skills. Keep practicing, and you'll become a composition function expert in no time!

Understanding the concepts of functions and how to manipulate them is fundamental to many mathematical principles. This includes not only algebra but also calculus and other advanced topics. Therefore, mastering the basic concepts is a great investment in your mathematical journey! Keep in mind to practice as much as possible. With enough practice, you will be able to solve even more complex problems. Never be afraid to look at different resources. Each one can explain things in a way that you can better understand them.