Domain Of Composite Function F(g(x)): Step-by-Step Solution

Hey math enthusiasts! Today, we're diving into the fascinating world of composite functions and figuring out how to pinpoint their domains. Specifically, we'll tackle a problem where we need to find the domain of $(f \circ g)(x)$, given that $f(x) = \frac{x-3}{x}$ and $g(x) = 5x - 4$. Don't worry, we'll break it down step by step so it's super easy to follow. So, grab your thinking caps, and let's get started!

Understanding Composite Functions

Before we jump into the nitty-gritty, let's quickly recap what composite functions are all about. Think of it like a machine where you feed in an input, and it goes through multiple stages of processing. In mathematical terms, a composite function is a function that is applied to the result of another function. The notation $(f \circ g)(x)$ means we first apply the function $g$ to $x$, and then we take the result and plug it into the function $f$. Essentially, it's $f(g(x))$. This seemingly simple concept can lead to some interesting challenges, especially when it comes to determining the domain.

The domain of a function, as you guys probably already know, is the set of all possible input values (x-values) that will produce a valid output. For rational functions (functions that are fractions with polynomials), like our $f(x)$ here, the domain is restricted by values that would make the denominator zero. We can't divide by zero, right? That's math's big no-no! Also, when we're dealing with composite functions, we need to consider the domain restrictions of both the inner function (in our case, $g(x)$) and the outer function (in our case, $f(x)$) after the composition.

Why is finding the domain so important? Well, it tells us where the function is actually defined. Imagine trying to bake a cake without knowing the correct temperature for your oven – you might end up with a disaster! Similarly, if we don't know the domain of a function, we might be trying to evaluate it at points where it simply doesn't exist. So, let’s avoid mathematical catastrophes and make sure we nail down the domain correctly!

Step 1: Find the Composite Function $(f ext{ ∘ } g)(x)$

Our first mission is to actually create the composite function. Remember, $(f \circ g)(x)$ means $f(g(x))$. So, we'll take the function $g(x)$ and plug it into $f(x)$ wherever we see an $x$.

We have:

• $f(x) = \frac{x-3}{x}$
• $g(x) = 5x - 4$

So, to find $f(g(x))$, we substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(5x - 4) = \frac{(5x - 4) - 3}{5x - 4}$

Now, let's simplify this expression:

$f(g(x)) = \frac{5x - 4 - 3}{5x - 4} = \frac{5x - 7}{5x - 4}$

Alright, we've got our composite function! It looks like a rational function, which means we need to be extra careful about the denominator. This is where our domain restrictions will come into play. We're one step closer, guys!

Step 2: Identify Potential Domain Restrictions

Okay, now that we have the composite function, $\frac{5x - 7}{5x - 4}$, we need to think about potential problems. Remember our golden rule: the denominator of a fraction cannot be zero. So, we need to figure out what values of $x$ would make the denominator, $5x - 4$, equal to zero. These values will be excluded from our domain.

Let's set the denominator equal to zero and solve for $x$:

$5x - 4 = 0$

Add 4 to both sides:

$5x = 4$

Divide both sides by 5:

$x = \frac{4}{5}$

So, we've found a critical value! If $x$ is equal to $\frac{4}{5}$, the denominator of our composite function becomes zero, and the function is undefined. This means $\frac{4}{5}$ cannot be part of our domain. But hold on, we're not done yet! There's another potential restriction we need to consider.

We also need to think about the domain of the inner function, $g(x)$. In this case, $g(x) = 5x - 4$. This is a linear function, and linear functions are defined for all real numbers. So, there are no domain restrictions coming from $g(x)$ itself. However, it's always a good practice to check! Sometimes the inner function can have its own limitations, which would then affect the overall domain of the composite function. Good job on remembering to check this, guys!

Step 3: Express the Domain in Set Notation

We've identified that $x$ cannot be equal to $\frac{4}{5}$. Now, we need to express this mathematically. The domain is the set of all real numbers except for $\frac{4}{5}$. There are a couple of ways we can write this using set notation.

One way is to use the set-builder notation:

$\left\{x \left\vert\, x \neq \frac{4}{5}\right.\right\}$

This notation reads as "the set of all $x$ such that $x$ is not equal to $\frac{4}{5}$." It's a concise and precise way to describe our domain.

Another way to express the domain is using interval notation. We can represent all real numbers less than $\frac{4}{5}$ as the interval $(-\infty, \frac{4}{5})$ and all real numbers greater than $\frac{4}{5}$ as the interval $(\frac{4}{5}, \infty)$. To combine these intervals, we use the union symbol, $\cup$:

$(-\infty, \frac{4}{5}) \cup (\frac{4}{5}, \infty)$

This notation means the domain includes all numbers from negative infinity up to (but not including) $\frac{4}{5}$, as well as all numbers from $\frac{4}{5}$ (not including) to positive infinity.

Both of these notations accurately represent the domain of our composite function. You can choose whichever one you find more clear and convenient! In the context of multiple-choice questions, you'll often see the set-builder notation used, so it's good to be familiar with it. You guys are doing great so far!

Step 4: State the Final Answer

We've done all the hard work! We found the composite function, identified the domain restrictions, and expressed the domain in set notation. Now, let's put it all together and state our final answer.

The domain of $(f \circ g)(x)$ is $\left\{x \left\vert\, x \neq \frac{4}{5}\right.\right\}$.

That's it! We've successfully navigated the world of composite functions and domains. Give yourselves a pat on the back!

Looking back at the multiple-choice options, the correct answer is C: $\left\{x \left\vert\, x \neq \frac{4}{5}\right.\right\}$

Remember, the key to finding the domain of a composite function is to consider the domain restrictions of both the inner and outer functions. Don't forget to check for values that would make the denominator zero or cause other mathematical issues (like taking the square root of a negative number). You guys got this!

Practice Makes Perfect

Now that we've walked through this problem together, the best way to solidify your understanding is to practice! Try tackling similar problems on your own. You can change the functions $f(x)$ and $g(x)$ and see how the domain of the composite function changes. Experiment with different types of functions, like square roots or logarithms, to add an extra layer of challenge. The more you practice, the more confident you'll become in finding the domains of composite functions.

And hey, if you ever get stuck, don't hesitate to ask for help! Math is a team sport, and we're all in this together. You can always revisit this guide, check out other resources online, or ask your teacher or classmates for clarification. Remember, the goal is to understand the concepts, not just memorize the steps. Keep up the great work, guys, and you'll be mastering composite functions in no time!

Conclusion

So, there you have it! Finding the domain of a composite function might seem a bit daunting at first, but by breaking it down into clear, manageable steps, it becomes much more approachable. We've learned how to find the composite function, identify potential domain restrictions, and express the domain in set notation. Remember to always consider the domains of both the inner and outer functions, and don't be afraid to practice! You've got the tools and the knowledge to conquer these types of problems. Keep exploring, keep learning, and most importantly, keep having fun with math! You guys are awesome, and I'm excited to see what mathematical challenges you'll tackle next. Until then, happy calculating!