Function Composition Problems And Solutions F(x) And G(x)
Hey guys! Let's dive into some cool problems involving function composition. If you're scratching your head over what F ∘ g (x) or g ∘ F (x) actually means, don't worry, we'll break it down step-by-step. We've got some examples here with F(x) = x^2 + 8x + 5 and g(x) = 3x + 8, and we're going to figure out how to solve for different compositions and values. So, grab your pencils, and let's get started!
Understanding Function Composition
Before we jump into solving, let's quickly recap what function composition really means. Think of it like this: you're plugging one function into another. When you see F ∘ g (x), it means you're taking the function g(x) and plugging it into F(x). It's like a chain reaction – the output of g(x) becomes the input of F(x). Similarly, g ∘ F (x) means you're plugging F(x) into g(x). Getting this concept down is super crucial, so make sure it clicks before we move on.
Key Concepts in Function Composition
To really master function composition, it’s essential to understand a few key concepts. At its core, function composition involves combining two functions by using the output of one function as the input for another. This process can be visualized as a sequence of operations where the first function transforms the initial input, and the second function further transforms the result. The notation F ∘ g (x) represents the composition of F with g, meaning we first apply g to x, and then apply F to the result. This is a crucial point to remember: the order matters! F ∘ g (x) is generally not the same as g ∘ F (x).
Another vital aspect of function composition is understanding the domain and range of the resulting composite function. 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, you need to ensure that the output of g(x) is a valid input for F(x). This often involves checking for restrictions such as division by zero or square roots of negative numbers. Similarly, the range of the composite function will depend on the ranges of both F and g, and how they interact with each other.
Moreover, function composition is not just a mathematical curiosity; it has practical applications in various fields. In computer science, for example, it is used to build complex algorithms by combining simpler functions. In physics, it can be used to model systems where multiple processes occur sequentially. Grasping these underlying concepts will not only help you solve problems but also appreciate the broader significance of function composition.
Problem Breakdown: F(x) = x^2 + 8x + 5 and g(x) = 3x + 8
Okay, let's get to the heart of the matter. We're given two functions: F(x) = x^2 + 8x + 5 and g(x) = 3x + 8. Our mission is to find:
a. F ∘ g (x) b. F ∘ g (1/2) c. g ∘ F (x) d. g ∘ F (-3)
We'll tackle each one step by step, so you can see exactly how it's done.
Step-by-Step Solution: Part A - F ∘ g (x)
First up, let's find F ∘ g (x). Remember, this means we're plugging g(x) into F(x). So, wherever we see an 'x' in F(x), we're going to replace it with the entire function g(x), which is 3x + 8. Let's write it out:
F(g(x)) = F(3x + 8) = (3x + 8)^2 + 8(3x + 8) + 5
Now, we need to expand and simplify this expression. The first part, (3x + 8)^2, can be expanded as (3x + 8)(3x + 8). Using the good ol' FOIL method (First, Outer, Inner, Last), we get:
(3x + 8)(3x + 8) = 9x^2 + 24x + 24x + 64 = 9x^2 + 48x + 64
Next, let's distribute the 8 in the second term:
8(3x + 8) = 24x + 64
Now, we can put it all together:
F(g(x)) = 9x^2 + 48x + 64 + 24x + 64 + 5
Finally, let's combine like terms:
F(g(x)) = 9x^2 + (48x + 24x) + (64 + 64 + 5) = 9x^2 + 72x + 133
So, F ∘ g (x) = 9x^2 + 72x + 133. Great job, we've nailed the first one!
Deeper Dive into F ∘ g (x)
To truly understand the result we obtained for F ∘ g (x), it’s beneficial to break down the process and analyze each step. We started by recognizing that F ∘ g (x) means applying the function g to x first, and then applying the function F to the result. This sequential application is the essence of function composition. Substituting g(x) = 3x + 8 into F(x) = x^2 + 8x + 5 gave us F(3x + 8) = (3x + 8)^2 + 8(3x + 8) + 5.
The next critical step was expanding the expression. The term (3x + 8)^2 required careful application of the distributive property or the FOIL method, which we correctly computed as 9x^2 + 48x + 64. Distributing the 8 in the term 8(3x + 8) yielded 24x + 64. Combining these results with the constant term 5, we obtained the expression 9x^2 + 48x + 64 + 24x + 64 + 5.
Finally, we simplified the expression by combining like terms. The x^2 term remained as 9x^2. The x terms, 48x and 24x, combined to give 72x. The constant terms, 64, 64, and 5, summed up to 133. Thus, we arrived at the final expression F ∘ g (x) = 9x^2 + 72x + 133. This quadratic function represents the composition of F with g. This thorough breakdown not only confirms our solution but also enhances our understanding of the process involved in function composition.
Step-by-Step Solution: Part B - F ∘ g (1/2)
Now, let's tackle F ∘ g (1/2). Guess what? We've already done the hard part! We know that F ∘ g (x) = 9x^2 + 72x + 133. So, to find F ∘ g (1/2), all we need to do is plug in x = 1/2 into this expression. Let's do it:
F ∘ g (1/2) = 9(1/2)^2 + 72(1/2) + 133
First, let's square 1/2:
(1/2)^2 = 1/4
Now, plug that back in:
F ∘ g (1/2) = 9(1/4) + 72(1/2) + 133
Next, let's multiply:
9(1/4) = 9/4 72(1/2) = 36
So, we have:
F ∘ g (1/2) = 9/4 + 36 + 133
To add these, we need a common denominator. Let's convert 36 and 133 to fractions with a denominator of 4:
36 = 36 * 4 / 4 = 144/4 133 = 133 * 4 / 4 = 532/4
Now we can add:
F ∘ g (1/2) = 9/4 + 144/4 + 532/4 = (9 + 144 + 532) / 4 = 685/4
So, F ∘ g (1/2) = 685/4. You can also write this as a mixed number or a decimal if you like, but the fraction is perfectly fine!
In-depth Look at F ∘ g (1/2)
To gain a deeper understanding of our solution for F ∘ g (1/2), it's valuable to revisit the step-by-step calculation. We started with our previously derived composite function, F ∘ g (x) = 9x^2 + 72x + 133, and substituted x = 1/2 into the expression. This substitution is a fundamental aspect of evaluating functions at specific points.
The next step involved squaring 1/2, which resulted in (1/2)^2 = 1/4. We then substituted this value back into our equation, obtaining F ∘ g (1/2) = 9(1/4) + 72(1/2) + 133. Each term was then evaluated separately. Multiplying 9 by 1/4 gave us 9/4, and multiplying 72 by 1/2 yielded 36. At this point, our equation looked like F ∘ g (1/2) = 9/4 + 36 + 133.
To add these terms, we needed a common denominator. Converting 36 and 133 to fractions with a denominator of 4, we found that 36 = 144/4 and 133 = 532/4. Our equation then became F ∘ g (1/2) = 9/4 + 144/4 + 532/4. Adding the numerators together, we got 9 + 144 + 532 = 685. Thus, F ∘ g (1/2) = 685/4. This result is an exact fractional representation, which can also be expressed as a mixed number (171 1/4) or a decimal (171.25). By meticulously reviewing each step, we reinforce our understanding of evaluating composite functions.
Step-by-Step Solution: Part C - g ∘ F (x)
Alright, let's switch gears and find g ∘ F (x). This time, we're plugging F(x) into g(x). So, wherever we see an 'x' in g(x), we're going to replace it with the entire function F(x), which is x^2 + 8x + 5. Let's write it out:
g(F(x)) = g(x^2 + 8x + 5) = 3(x^2 + 8x + 5) + 8
Now, we just need to distribute the 3:
3(x^2 + 8x + 5) = 3x^2 + 24x + 15
So, we have:
g(F(x)) = 3x^2 + 24x + 15 + 8
Finally, let's combine like terms:
g(F(x)) = 3x^2 + 24x + (15 + 8) = 3x^2 + 24x + 23
Therefore, g ∘ F (x) = 3x^2 + 24x + 23. See? It's just about plugging and chugging!
Detailed Explanation of g ∘ F (x)
To fully grasp the computation of g ∘ F (x), let’s dissect the process. The key concept here is that we are composing the function g with F, which means we are using the output of F(x) as the input for g. Given g(x) = 3x + 8 and F(x) = x^2 + 8x + 5, we need to find g(F(x)), which involves substituting F(x) into g(x).
We start by replacing the 'x' in g(x) with the entire function F(x): g(F(x)) = 3(x^2 + 8x + 5) + 8. The next step is to distribute the 3 across the terms inside the parentheses: 3(x^2 + 8x + 5) = 3x^2 + 24x + 15. This distribution is a critical algebraic manipulation that ensures we correctly apply the function g to every part of F(x).
After distributing, we have g(F(x)) = 3x^2 + 24x + 15 + 8. The final step is to combine like terms. In this case, we only need to add the constant terms: 15 + 8 = 23. This gives us the simplified expression g ∘ F (x) = 3x^2 + 24x + 23. This quadratic function represents the composition of g with F. Breaking down each step in this manner clarifies the process and reinforces the understanding of function composition.
Step-by-Step Solution: Part D - g ∘ F (-3)
Last but not least, let's find g ∘ F (-3). Just like before, we've already done the heavy lifting. We know that g ∘ F (x) = 3x^2 + 24x + 23. To find g ∘ F (-3), we simply plug in x = -3:
g ∘ F (-3) = 3(-3)^2 + 24(-3) + 23
First, let's square -3:
(-3)^2 = 9
Now, plug that back in:
g ∘ F (-3) = 3(9) + 24(-3) + 23
Next, let's multiply:
3(9) = 27 24(-3) = -72
So, we have:
g ∘ F (-3) = 27 - 72 + 23
Now, let's add and subtract:
g ∘ F (-3) = (27 + 23) - 72 = 50 - 72 = -22
Therefore, g ∘ F (-3) = -22. We've conquered all parts of the problem! Give yourself a pat on the back!
Deconstructing the Solution for g ∘ F (-3)
To thoroughly understand the solution for g ∘ F (-3), let’s revisit each step of the calculation. We started with our composite function g ∘ F (x) = 3x^2 + 24x + 23, which we derived earlier. The task was to evaluate this composite function at x = -3. This involves substituting -3 for x in the expression, which is a fundamental procedure in function evaluation.
The first step in the substitution was to replace x with -3: g ∘ F (-3) = 3(-3)^2 + 24(-3) + 23. Next, we simplified the terms according to the order of operations. Squaring -3 gives us (-3)^2 = 9. Substituting this back into the expression, we have g ∘ F (-3) = 3(9) + 24(-3) + 23.
Then, we performed the multiplications: 3(9) = 27 and 24(-3) = -72. This resulted in the expression g ∘ F (-3) = 27 - 72 + 23. Finally, we performed the addition and subtraction from left to right: 27 - 72 = -45, and -45 + 23 = -22. Thus, we arrived at the final answer g ∘ F (-3) = -22. By carefully examining each step, we solidify our understanding of how to evaluate composite functions at specific values.
Final Thoughts
So there you have it! We've successfully navigated the world of function composition, solving for F ∘ g (x), F ∘ g (1/2), g ∘ F (x), and g ∘ F (-3). The key takeaway here is to remember what the notation means and to take it one step at a time. Plug the inner function into the outer function, simplify, and you're golden!
Remember, practice makes perfect. The more you work through these types of problems, the more comfortable you'll become. So, keep at it, and you'll be a function composition pro in no time!
Tips for Mastering Function Composition
To truly master function composition, here are some handy tips that can help you along the way. First and foremost, always remember the order of operations. Function composition involves applying functions in a specific sequence, so it's crucial to apply the inner function first and then the outer function. Writing out the steps can help prevent errors and keep the process clear in your mind.
Another essential tip is to simplify each function as much as possible before composing them. This can make the algebra much easier and reduce the chances of making mistakes. For example, if you have a complex function like F(x) = (2x^2 + 4x) / 2, simplify it to F(x) = x^2 + 2x before plugging it into another function. This will save you a lot of time and effort.
Don't hesitate to use parentheses liberally. Parentheses help to clearly define the order of operations and prevent confusion. When substituting one function into another, enclose the entire function in parentheses to ensure it is treated as a single entity. This is particularly important when dealing with more complex expressions.
Lastly, practice with a variety of examples. The more you practice, the more comfortable you will become with function composition. Start with simpler functions and gradually work your way up to more challenging ones. Look for patterns and common techniques that can help you solve problems more efficiently. With consistent effort and these tips in mind, you’ll be well on your way to mastering function composition.
