Composite Functions Of F(x) = X/(x-9) And G(x) = -8/x And Their Domains
Hey guys! Today, we're diving deep into the fascinating world of composite functions. We'll be working with two specific functions: f(x) = x/(x-9) and g(x) = -8/x. Our mission? To find the composite functions (f ∘ g)(x), (g ∘ f)(x), (f ∘ f)(x), and (g ∘ g)(x), and most importantly, to determine the domain of each. So, buckle up and let's get started!
(a) Finding (f ∘ g)(x) and Its Domain
First, let's tackle (f ∘ g)(x), which reads as "f of g of x." What does this actually mean? It means we're going to take the function g(x) and plug it into the function f(x). Think of it like a mathematical Russian doll! So, wherever we see an x in f(x), we're going to replace it with the entire expression for g(x).
Our functions are:
- f(x) = x / (x - 9)
- g(x) = -8 / x
Let's do the substitution:
(f ∘ g)(x) = f(g(x)) = f(-8/x) = (-8/x) / ((-8/x) - 9)
Okay, that looks a little messy, right? Let's simplify this complex fraction. To do that, we'll multiply both the numerator and the denominator by x. This will clear out the fractions within the fraction:
(f ∘ g)(x) = [(-8/x) * x] / [((-8/x) - 9) * x] = -8 / (-8 - 9x)
We can even make it look a bit cleaner by factoring out a -1 from the denominator:
(f ∘ g)(x) = -8 / (-1(8 + 9x)) = 8 / (8 + 9x)
Alright, we've found our composite function: (f ∘ g)(x) = 8 / (8 + 9x). But we're not done yet! We need to determine the domain of this function.
The domain, in simple terms, is the set of all possible input values (x-values) that will give us a valid output. For rational functions (fractions with x in the denominator), we need to watch out for values that make the denominator equal to zero. Why? Because division by zero is a big no-no in the math world – it's undefined!
So, let's find those pesky values that make the denominator zero:
8 + 9x = 0
9x = -8
x = -8/9
This means that x = -8/9 is the one value we need to exclude from our domain. Also, we need to consider the domain of the inner function, g(x) = -8/x. Here, x cannot be 0. Therefore, the domain of (f ∘ g)(x) is all real numbers except for -8/9 and 0. We can express this in interval notation as:
Domain of (f ∘ g)(x): (-∞, -8/9) ∪ (-8/9, 0) ∪ (0, ∞)
Remember, when finding the domain of a composite function, always consider the domain restrictions of both the inner function and the resulting composite function.
(b) Finding (g ∘ f)(x) and Its Domain
Now, let's switch things up and find (g ∘ f)(x), which means "g of f of x." This time, we're plugging f(x) into g(x). Get ready for some more mathematical gymnastics!
Our functions, as a reminder, are:
- f(x) = x / (x - 9)
- g(x) = -8 / x
Let's substitute f(x) into g(x):
(g ∘ f)(x) = g(f(x)) = g(x / (x - 9)) = -8 / (x / (x - 9))
This looks like another complex fraction, but don't worry, we've got this! To simplify, we can multiply the numerator and denominator by the denominator of the inner fraction, which is (x - 9):
(g ∘ f)(x) = [-8 * (x - 9)] / [(x / (x - 9)) * (x - 9)] = -8(x - 9) / x
Distributing the -8 in the numerator, we get:
(g ∘ f)(x) = (-8x + 72) / x
So, our composite function is (g ∘ f)(x) = (-8x + 72) / x. Time to find its domain!
Again, we need to watch out for values that make the denominator zero. In this case, it's pretty straightforward: x cannot be 0.
But hold on! We also need to consider the domain of the inner function, f(x) = x / (x - 9). Here, x cannot be 9, because that would make the denominator zero.
Therefore, the domain of (g ∘ f)(x) is all real numbers except for 0 and 9. In interval notation, this is:
Domain of (g ∘ f)(x): (-∞, 0) ∪ (0, 9) ∪ (9, ∞)
Key takeaway: Always double-check for domain restrictions from both the inner function and the simplified composite function.
(c) Finding (f ∘ f)(x) and Its Domain
Now, let's get a little meta! We're going to find (f ∘ f)(x), which means "f of f of x." This time, we're plugging the function f(x) back into itself. Sounds like fun, right?
Our function f(x) is:
- f(x) = x / (x - 9)
Let's do the substitution:
(f ∘ f)(x) = f(f(x)) = f(x / (x - 9)) = (x / (x - 9)) / ((x / (x - 9)) - 9)
Oh boy, another complex fraction! Let's simplify by multiplying the numerator and denominator by (x - 9):
(f ∘ f)(x) = [(x / (x - 9)) * (x - 9)] / [((x / (x - 9)) - 9) * (x - 9)] = x / (x - 9(x - 9))
Distributing the -9 in the denominator, we get:
(f ∘ f)(x) = x / (x - 9x + 81) = x / (-8x + 81)
So, our composite function is (f ∘ f)(x) = x / (-8x + 81). Let's find that domain!
We need to find the values that make the denominator zero:
-8x + 81 = 0
-8x = -81
x = 81/8
So, x = 81/8 is one value we need to exclude. But remember, we also need to consider the domain of the inner function, which is also f(x) = x / (x - 9). This means x cannot be 9.
Therefore, the domain of (f ∘ f)(x) is all real numbers except for 9 and 81/8. In interval notation:
Domain of (f ∘ f)(x): (-∞, 9) ∪ (9, 81/8) ∪ (81/8, ∞)
Pro Tip: When composing a function with itself, make sure to carefully consider the domain restrictions at each step.
(d) Finding (g ∘ g)(x) and Its Domain
Last but not least, let's find (g ∘ g)(x), which means "g of g of x." We're plugging g(x) back into itself. This should be fun!
Our function g(x) is:
- g(x) = -8 / x
Let's substitute:
(g ∘ g)(x) = g(g(x)) = g(-8/x) = -8 / (-8/x)
This looks simpler than the previous ones! To simplify, we can multiply the numerator and denominator by x:
(g ∘ g)(x) = (-8 * x) / ((-8/x) * x) = -8x / -8 = x
Wow! Our composite function simplifies to (g ∘ g)(x) = x. That's pretty neat!
Now, let's find the domain. At first glance, it might seem like the domain is all real numbers since our simplified function is just x. However, we need to be super careful and remember the domain of the inner function, which is g(x) = -8/x. This means x cannot be 0.
Therefore, even though the simplified composite function is x, the domain of (g ∘ g)(x) is all real numbers except for 0. In interval notation:
Domain of (g ∘ g)(x): (-∞, 0) ∪ (0, ∞)
Important Reminder: Even if a composite function simplifies to something very simple, always go back and check the domain restrictions of the original functions!
Conclusion
And there you have it, folks! We've successfully found the composite functions (f ∘ g)(x), (g ∘ f)(x), (f ∘ f)(x), and (g ∘ g)(x), and we've carefully determined the domain of each. Remember, the key to mastering composite functions is to take it step by step, pay attention to the order of operations, and always, always consider the domain restrictions. Keep practicing, and you'll be a composite function pro in no time!