Composite Functions For F(x) = 4x - 9 And G(x) = X^2 Domain Analysis

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of composite functions. We'll be working with two specific functions: f(x) = 4x - 9 and g(x) = x^2. Our mission? To find the composite functions f ∘ g, g ∘ f, f ∘ f, and g ∘ g, and most importantly, to determine the domain of each resulting composite function. So, buckle up and let's get started!

(a) Delving into f ∘ g: The Magic of Function Composition

Let's kick things off with the composite function f ∘ g, which is read as "f of g." What does this actually mean, you ask? Well, it means we're going to take the function g(x) and plug it into the function f(x). Think of it as a function within a function – a mathematical Matryoshka doll, if you will!

So, how do we do it? Remember that g(x) = x^2. We'll substitute x^2 wherever we see x in the function f(x) = 4x - 9. This gives us:

(f ∘ g)(x) = f(g(x)) = f(x^2) = 4(x^2) - 9 = 4x^2 - 9

There you have it! The composite function (f ∘ g)(x) is 4x^2 - 9. Easy peasy, right?

But our journey doesn't end here. We need to determine the domain of this new function. Now, the domain of a function is the set of all possible input values (x-values) that will produce a valid output. For polynomial functions like 4x^2 - 9, there are no restrictions on the input. We can plug in any real number, and we'll get a real number output. Therefore, the domain of (f ∘ g)(x) is all real numbers. We can express this mathematically as:

Domain of (f ∘ g)(x): (-∞, ∞)

Key Points for f ∘ g(x):

• Understanding Composition: f ∘ g means applying g first, then applying f to the result.
• Substitution is Key: Substitute g(x) into f(x).
• Domain Consideration: For polynomials, the domain is generally all real numbers.
• Resultant Function: The composite function (f ∘ g)(x) simplifies to 4x^2 - 9.

(b) Unraveling g ∘ f: Switching the Order

Now, let's switch things up and explore the composite function g ∘ f, which is read as "g of f." This time, we're going to plug the function f(x) into the function g(x). Notice how the order matters here! Function composition is not always commutative, meaning f ∘ g is not necessarily the same as g ∘ f.

We know that f(x) = 4x - 9 and g(x) = x^2. So, we'll substitute (4x - 9) wherever we see x in the function g(x). This gives us:

(g ∘ f)(x) = g(f(x)) = g(4x - 9) = (4x - 9)^2

We can expand this expression to get:

(g ∘ f)(x) = (4x - 9)(4x - 9) = 16x^2 - 72x + 81

So, the composite function (g ∘ f)(x) is 16x^2 - 72x + 81. Again, we need to determine the domain. Since this is also a polynomial function, there are no restrictions on the input. We can plug in any real number and get a real number output. Therefore, the domain of (g ∘ f)(x) is all real numbers:

Domain of (g ∘ f)(x): (-∞, ∞)

Key Points for g ∘ f(x):

• Order Matters: g ∘ f is different from f ∘ g.
• Substitution Process: Substitute f(x) into g(x).
• Expanding the Result: Expand the squared term to get 16x^2 - 72x + 81.
• Domain Remains Unrestricted: The domain is still all real numbers.

(c) Exploring f ∘ f: Composing a Function with Itself

Let's get a little meta and explore the composite function f ∘ f. This means we're plugging the function f(x) into itself! It might seem a bit strange, but it's a perfectly valid operation.

We have f(x) = 4x - 9. So, we'll substitute (4x - 9) wherever we see x in the function f(x). This gives us:

(f ∘ f)(x) = f(f(x)) = f(4x - 9) = 4(4x - 9) - 9

Now, let's simplify this expression:

(f ∘ f)(x) = 16x - 36 - 9 = 16x - 45

So, the composite function (f ∘ f)(x) is 16x - 45. As with the previous examples, this is a linear function, which is a type of polynomial. There are no restrictions on the input, so the domain is all real numbers:

Domain of (f ∘ f)(x): (-∞, ∞)

Key Points for f ∘ f(x):

• Self-Composition: Applying the function to itself.
• Repeated Substitution: Substituting f(x) into f(x).
• Simplifying the Result: The composite function simplifies to 16x - 45.
• Domain Stability: The domain remains all real numbers.

(d) Diving into g ∘ g: Squaring a Square

Finally, let's tackle the composite function g ∘ g. This means we're plugging the function g(x) into itself. Since g(x) = x^2, we're essentially squaring a square!

Let's perform the substitution:

(g ∘ g)(x) = g(g(x)) = g(x^2) = (x²⁾2

Simplifying this expression, we get:

(g ∘ g)(x) = x^4

So, the composite function (g ∘ g)(x) is x^4. This is another polynomial function, and like the others, there are no restrictions on the input. Therefore, the domain is all real numbers:

Domain of (g ∘ g)(x): (-∞, ∞)

Key Points for g ∘ g(x):

• Squaring a Square: Applying the square function to itself.
• Power Rule: Using the power rule of exponents, (x²⁾2 simplifies to x^4.
• Polynomial Behavior: The result is a polynomial function.
• Consistent Domain: The domain is all real numbers.

Wrapping Up: Mastering Composite Functions

And there you have it, guys! We've successfully navigated the world of composite functions, finding f ∘ g, g ∘ f, f ∘ f, and g ∘ g for the given functions f(x) = 4x - 9 and g(x) = x^2. We've also carefully determined the domain of each composite function, which in all cases turned out to be all real numbers.

Remember the key takeaways:

• Function composition involves plugging one function into another.
• The order of composition matters (f ∘ g ≠ g ∘ f in general).
• The domain of a composite function is the set of all input values that produce a valid output.
• Polynomial functions generally have a domain of all real numbers.

I hope this comprehensive guide has helped you solidify your understanding of composite functions. Keep practicing, and you'll become a function composition master in no time!