Find H(1) Given Composite Function H(x) = (f∘g)(x)

Hey guys! Let's dive into a fun math problem today. We're going to figure out how to find the value of ${h(1)}$ when we know that ${h(x)}$ is a composite function made up of ${f(x)}$ and ${g(x)}$. This might sound a little complicated at first, but trust me, we'll break it down step by step so it's super easy to understand. So, let's get started and unravel this mathematical puzzle together! We are given the functions ${f(x) = x^2}$ and ${g(x) = 8x + 3}$, and we need to find ${h(1)}$ where ${h(x) = (f \circ g)(x)}$.

Understanding Composite Functions

Before we jump into solving the problem, let's quickly recap what a composite function is. Think of it like this: a composite function is a function that's plugged into another function. In our case, ${h(x) = (f \circ g)(x)}$ means we're plugging the function ${g(x)}$ into the function ${f(x)}$. Basically, we first apply ${g}$ to ${x}$, and then we apply ${f}$ to the result. It's like a mathematical assembly line! The notation ${(f \circ g)(x)}$ is just a fancy way of writing ${f(g(x))}$. They both mean the same thing. So, if you see either of these notations, don't let them intimidate you. Just remember that we're dealing with one function nestled inside another.

To really nail this down, let's think of a simple example. Imagine ${f(x) = x + 1}$ and ${g(x) = 2x}$. To find ${(f \circ g)(x)}$, we would first find ${g(x)}$, which is ${2x}$. Then, we plug this result into ${f(x)}$. So, ${f(g(x))}$ would be ${f(2x) = 2x + 1}$. See? It's all about substituting one function into another. Understanding this concept is crucial for tackling problems like the one we have today. With this foundation in place, we're ready to dive into the specifics of our problem and find ${h(1)}$. Remember, math is like building blocks – each concept builds on the previous one. So, let's keep building our understanding and solve this thing!

Step-by-Step Solution to find h(1)

Okay, let's get down to business and solve for ${h(1)}$. Remember, we have ${f(x) = x^2}$, ${g(x) = 8x + 3}$, and ${h(x) = (f \circ g)(x)}$. Our mission is to find the value of ${h(1)}$. To do this, we'll follow a simple two-step process. First, we'll find the expression for ${h(x)}$, and then we'll substitute ${x = 1}$ into that expression. It's like a treasure hunt – we need to find the map (the expression for ${h(x)}$) before we can find the treasure (the value of ${h(1)}$).

Step 1: Find the expression for h(x)

As we discussed earlier, ${h(x) = (f \circ g)(x)}$ is the same as ${f(g(x))}$. This means we need to plug ${g(x)}$ into ${f(x)}$. Let's do it! We know that ${g(x) = 8x + 3}$. So, we need to substitute ${8x + 3}$ into ${f(x)}$. Remember, ${f(x) = x^2}$. This means we're going to replace the ${x}$ in ${f(x)}$ with ${8x + 3}$. So, we get:

${ h(x) = f(g(x)) = f(8x + 3) = (8x + 3)^2 }$

Awesome! We've found the expression for ${h(x)}$. It's ${(8x + 3)^2}$. Now, we're one step closer to finding our treasure. The next part is straightforward. All we need to do is substitute ${x = 1}$ into this expression. Think of it like following the last instruction on our map – we're almost there!

Step 2: Substitute x = 1 into h(x)

Now that we have ${h(x) = (8x + 3)^2}$, let's substitute ${x = 1}$ to find ${h(1)}$. This is where the magic happens! We simply replace every ${x}$ in the expression with ${1}$. So, we get:

${ h(1) = (8(1) + 3)^2 }$

Now, let's simplify this. First, we do the multiplication inside the parentheses:

${ h(1) = (8 + 3)^2 }$

Then, we add the numbers inside the parentheses:

${ h(1) = (11)^2 }$

Finally, we square ${11}$:

${ h(1) = 121 }$

Boom! We've found it! The value of ${h(1)}$ is ${121}$. That wasn't so bad, was it? We took a seemingly complex problem and broke it down into manageable steps. This is a powerful strategy in math – and in life! Whenever you face a tough challenge, try breaking it down into smaller parts. You might be surprised at how easy it becomes.

Alternative Method: Evaluating g(1) First

Alright, guys, there's another way we can tackle this problem, and it's always cool to have different tools in our mathematical toolbox! Instead of finding the general expression for ${h(x)}$ first, we can directly find ${g(1)}$ and then plug that value into ${f(x)}$. This method can be particularly handy when you only need to find the value of the composite function at a specific point, like ${x = 1}$ in our case. It's like taking a different route to the same destination – sometimes it might even be quicker!

Step 1: Evaluate g(1)

So, let's start by finding ${g(1)}$. We know that ${g(x) = 8x + 3}$. To find ${g(1)}$, we simply substitute ${x = 1}$ into this expression:

${ g(1) = 8(1) + 3 }$

Now, let's simplify:

${ g(1) = 8 + 3 }$

${ g(1) = 11 }$

Great! We found that ${g(1) = 11}$. This is a key piece of information. Now, we're ready to move on to the next step, where we'll use this value to find ${h(1)}$.

Step 2: Evaluate f(g(1))

Now that we know ${g(1) = 11}$, we can find ${h(1)}$ by evaluating ${f(g(1))}$, which is the same as ${f(11)}$. Remember, ${f(x) = x^2}$. So, to find ${f(11)}$, we substitute ${x = 11}$ into this expression:

${ f(11) = (11)^2 }$

Now, let's simplify:

${ f(11) = 121 }$

Ta-da! We've arrived at the same answer as before: ${h(1) = 121}$. See? Two different paths, same destination. This illustrates a beautiful aspect of mathematics – there's often more than one way to solve a problem. Choosing the method that feels most comfortable or efficient for you is part of the fun! This alternative method reinforces the concept of composite functions and gives you another tool to add to your problem-solving arsenal. Remember, the more techniques you know, the more confident you'll feel tackling any math challenge.

Common Mistakes to Avoid

Hey, before we wrap things up, let's chat about some common pitfalls that people often stumble into when dealing with composite functions. Knowing these mistakes can help you steer clear of them and ace your math problems! It's like knowing the traps on a treasure map – you'll be much more likely to find the treasure if you know where the dangers lie.

Mistake 1: Confusing the Order of Composition

One of the biggest slip-ups is mixing up the order of the functions in a composite function. Remember, ${(f \circ g)(x)}$ is not the same as ${(g \circ f)(x)}$. The order matters! ${(f \circ g)(x)}$ means we apply ${g}$ first, then ${f}$, while ${(g \circ f)(x)}$ means we apply ${f}$ first, then ${g}$. It's like putting on your socks and shoes – you gotta put your socks on first, or it's just weird! To avoid this, always pay close attention to the notation and remember that the function closest to the ${x}$ is the one you apply first.

For example, if we had calculated ${(g \circ f)(x)}$ instead of ${(f \circ g)(x)}$ in our problem, we would have gotten a completely different answer. This highlights how crucial it is to understand the order of operations in composite functions. Double-check yourself and make sure you're plugging the functions in the correct sequence.

Mistake 2: Incorrectly Substituting the Function

Another common mistake is messing up the substitution process. When you're plugging one function into another, it's super important to replace every instance of ${x}$ in the outer function with the entire inner function. Don't leave out any ${x}$'s, and make sure you use parentheses to keep things organized. This is especially important when the inner function has multiple terms.

In our problem, we had to substitute ${g(x) = 8x + 3}$ into ${f(x) = x^2}$. If we had forgotten the parentheses and written ${8x + 3^2}$ instead of ${(8x + 3)^2}$, we would have made a big mistake. The parentheses tell us that the entire expression ${8x + 3}$ is being squared, not just the ${3}$. So, always double-check your substitutions and make sure you're using parentheses when necessary. It's a small detail that can make a huge difference in your final answer.

Mistake 3: Not Simplifying Correctly

Even if you set up the problem correctly, you can still go wrong if you don't simplify the expression carefully. Remember to follow the order of operations (PEMDAS/BODMAS) and be mindful of your arithmetic. Simple errors in addition, subtraction, multiplication, or division can throw off your entire answer.

In our problem, we had to simplify ${(8(1) + 3)^2}$. If we had made a mistake in the addition or squaring, we would have ended up with the wrong value for ${h(1)}$. So, take your time, double-check your work, and don't rush through the simplification process. It's better to be accurate than fast. By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering composite functions and acing your math exams!

Conclusion

Alright, guys, we've reached the end of our mathematical journey for today! We successfully found the value of ${h(1)}$ given the composite function ${h(x) = (f \circ g)(x)}$, where ${f(x) = x^2}$ and ${g(x) = 8x + 3}$. We explored the concept of composite functions, worked through a step-by-step solution, discovered an alternative method, and even learned about common mistakes to avoid. That's a lot of ground covered! The key takeaway here is that composite functions, while they might seem intimidating at first, are actually quite manageable when you break them down into smaller, logical steps. Remember, it's all about plugging one function into another, and paying close attention to the order of operations.

Whether you choose to find the general expression for ${h(x)}$ first or evaluate ${g(1)}$ and then plug it into ${f(x)}$, the important thing is to understand the underlying concept and apply it correctly. And don't forget those parentheses! They're your friends when it comes to avoiding substitution errors. By practicing these types of problems and being mindful of the common mistakes, you'll build your confidence and become a composite function pro in no time. So, keep practicing, keep exploring, and keep having fun with math! You've got this!