Composite Functions: F(x)=x² & G(x)=x²+4 Explained

Hey guys! Today, we're diving into the fascinating world of composite functions. Think of it like a mathematical assembly line, where the output of one function becomes the input of another. We'll be exploring this concept using two simple yet powerful functions: f(x) = x² and g(x) = x² + 4. We'll break down how to find composite functions like f(g(x)), g(f(x)), f(f(x)), and g(g(x)), and most importantly, we'll figure out the domains of these new functions. Domains, in case you're wondering, are just the set of all possible input values (x-values) that a function can handle without blowing up or doing anything weird (like dividing by zero or taking the square root of a negative number).

So, grab your thinking caps, and let's get started!

(a) f o g: Composing f with g – f(g(x))

Let's start with the first composite function: f o g, which is read as "f composed with g." What this really means is f(g(x)). We're going to take the entire function g(x) and plug it in as the input for function f. It's like g(x) is a package that we're delivering to the function f. To find f(g(x)), the first key step involves substituting the expression for g(x) into f(x). Remember, g(x) = x² + 4 and f(x) = x². Wherever we see an 'x' in f(x), we're going to replace it with the entire expression for g(x), which is x² + 4. This gives us f(g(x)) = f(x² + 4) = (x² + 4)². Now, the next step is where the algebra magic happens. We need to expand the expression (x² + 4)². This means multiplying (x² + 4) by itself: (x² + 4)(x² + 4). Using the good old FOIL method (First, Outer, Inner, Last) or the distributive property, we get:

(x² + 4)(x² + 4) = x⁴ + 4x² + 4x² + 16. Combining like terms, the simplified result we're looking for is (f o g)(x) = x⁴ + 8x² + 16. This is our composite function, but we're not done yet! We still need to figure out the domain. The domain of a composite function is a bit tricky because we need to consider the domains of both the inner function (g(x) in this case) and the resulting composite function. First, let's think about g(x) = x² + 4. Are there any values of x that we can't plug into this function? Nope! We can square any real number and add 4, no problem. So, the domain of g(x) is all real numbers. Next, let's look at our composite function, (f o g)(x) = x⁴ + 8x² + 16. Again, are there any restrictions on the values of x we can use? We can raise any real number to the fourth power, multiply it by 8, and add 16. So, the domain of (f o g)(x) is also all real numbers. Since both g(x) and (f o g)(x) have domains of all real numbers, the domain of f o g is all real numbers. We can express this mathematically using interval notation as (-∞, ∞).

(b) g o f: Composing g with f – g(f(x))

Alright, let's switch things up and look at the composite function g o f, which is read as "g composed with f." This time, we're doing g(f(x)), meaning we're going to plug the entire function f(x) into the function g. So, f(x) is now our package being delivered to g(x). We'll start with substituting the expression for f(x) into g(x). Remember, f(x) = x² and g(x) = x² + 4. We'll replace the 'x' in g(x) with the entire expression for f(x), which is x². This gives us g(f(x)) = g(x²) = (x²)² + 4. Now we need to simplify the expression (x²)² + 4. The key here is understanding how exponents work. When we raise a power to a power, we multiply the exponents. So, (x²)² becomes x⁴. This simplifies our expression to (g o f)(x) = x⁴ + 4. Awesome! We have our composite function. But just like before, we need to consider the domain. We need to think about the domains of both the inner function, f(x), and the composite function, (g o f)(x). First, let's consider f(x) = x². Can we plug any real number into this function? Absolutely! We can square any real number, so the domain of f(x) is all real numbers. Now, let's look at the composite function (g o f)(x) = x⁴ + 4. Are there any restrictions on the values of x we can use here? We can raise any real number to the fourth power and add 4. So, the domain of (g o f)(x) is also all real numbers. Since both f(x) and (g o f)(x) have domains of all real numbers, the domain of g o f is all real numbers, or (-∞, ∞) in interval notation.

(c) f o f: Composing f with itself – f(f(x))

Now for something a little different: f o f, which is "f composed with f." This means we're plugging the function f(x) into itself! Think of it like a function looking in a mirror and seeing another version of itself. We're finding f(f(x)), meaning we substitute the expression for f(x) into itself. Since f(x) = x², we'll replace the 'x' in f(x) with x²: f(f(x)) = f(x²) = (x²)². Simplifying this expression is pretty straightforward. Remember the rule for exponents: when raising a power to a power, we multiply the exponents. So, (x²)² becomes x⁴. This means (f o f)(x) = x⁴. Cool! We've got the composite function. Let's figure out the domain. We need to think about the domain of the inner function, f(x), and the composite function, (f o f)(x). We already know that the domain of f(x) = x² is all real numbers because we can square any real number. Now, let's look at (f o f)(x) = x⁴. Can we raise any real number to the fourth power? Yes, we can! So, the domain of (f o f)(x) is also all real numbers. Since both the inner and outer functions have domains of all real numbers, the domain of f o f is all real numbers, which is (-∞, ∞) in interval notation.

(d) g o g: Composing g with itself – g(g(x))

Last but not least, we have g o g, or "g composed with g." This is similar to f o f, but this time we're plugging g(x) into itself. We're finding g(g(x)), so we substitute the expression for g(x) into itself. Remember, g(x) = x² + 4. We'll replace the 'x' in g(x) with the entire expression x² + 4: g(g(x)) = g(x² + 4) = (x² + 4)² + 4. Now we need to simplify this expression. We've actually already done the hard part of this simplification when we found f o g! We know that (x² + 4)² expands to x⁴ + 8x² + 16. So, we can substitute that in: g(g(x)) = x⁴ + 8x² + 16 + 4. Combining like terms, we get (g o g)(x) = x⁴ + 8x² + 20. Fantastic! We have our final composite function. Time for the domain! We need to consider the domain of the inner function, g(x), and the composite function, (g o g)(x). We know that the domain of g(x) = x² + 4 is all real numbers because we can square any real number and add 4. Looking at (g o g)(x) = x⁴ + 8x² + 20, can we think of any restrictions on the values of x? Nope! We can raise any real number to the fourth power, multiply it by 8, add a multiple of its square, and then add 20. So, the domain of (g o g)(x) is also all real numbers. Since both the inner and outer functions have domains of all real numbers, the domain of g o g is all real numbers, or (-∞, ∞) in interval notation.

Conclusion: Mastering Composite Functions

So, there you have it! We've successfully navigated the world of composite functions using f(x) = x² and g(x) = x² + 4. We've seen how to find f(g(x)), g(f(x)), f(f(x)), and g(g(x)), and we've carefully considered the domain of each composite function. Remember, the key to composite functions is understanding that you're plugging one entire function into another. And when it comes to the domain, always think about the domains of both the inner function and the resulting composite function. You've nailed it, guys! Keep practicing, and you'll be a composite function pro in no time.