Composite Function (f ∘ G)(x) And Domain Explained

by Alex Johnson 51 views

Let's dive into the world of composite functions! This article will guide you through finding the composite function (f ∘ g)(x) and determining its domain, given the functions f(x) = x/(x-2) and g(x) = -1/x. We'll break down each step in a clear and easy-to-understand manner. So, grab your math hats, and let's get started!

Understanding Composite Functions

Before we jump into the specifics, let's quickly recap what composite functions are all about. A composite function is essentially a function within a function. It's like a mathematical nesting doll! We denote it as (f ∘ g)(x), which is read as "f of g of x." This means we first apply the function g to x, and then we take the result and plug it into the function f.

Why are composite functions important? Well, they pop up in various areas of mathematics and its applications. They help us model complex relationships by breaking them down into simpler steps. For example, in physics, you might use composite functions to describe the position of an object over time, where one function determines the velocity and another determines the position based on the velocity. In computer science, they're used in function composition, where smaller functions are combined to create larger, more complex programs. Understanding composite functions is a fundamental skill that opens doors to more advanced mathematical concepts and real-world applications.

When working with composite functions, the domain is a crucial consideration. The domain of (f ∘ g)(x) isn't just the set of all x-values that work in the final composite function; it also includes any restrictions imposed by the inner function, g(x). Think of it like this: if g(x) produces an output that can't be used as an input for f(x), then that x-value isn't in the domain of the composite function. Therefore, to find the domain of (f ∘ g)(x), we need to consider the domains of both g(x) and f(x), and make sure that the output of g(x) falls within the domain of f(x).

Finding the Composite Function (f ∘ g)(x)

Now, let's get our hands dirty and find the composite function (f ∘ g)(x) for our given functions: f(x) = x/(x-2) and g(x) = -1/x. Remember, (f ∘ g)(x) means we substitute g(x) into f(x) wherever we see an 'x'.

Here's how we do it step-by-step:

  1. Write down the definition of (f ∘ g)(x): (f ∘ g)(x) = f(g(x))
  2. Substitute g(x) into f(x): This means replacing every 'x' in f(x) with the expression for g(x), which is -1/x. So, we get:f(g(x)) = f(-1/x) = (-1/x) / ((-1/x) - 2)
  3. Simplify the expression: This is where our algebraic skills come into play. We need to simplify the complex fraction. To do this, we can multiply both the numerator and the denominator by the least common multiple (LCM) of the denominators within the fraction, which in this case is 'x'.f(-1/x) = [(-1/x) * x] / [((-1/x) - 2) * x] = -1 / (-1 - 2x)
  4. Further simplification: We can multiply both the numerator and denominator by -1 to get rid of the negative signs: -1 / (-1 - 2x) = 1 / (1 + 2x)

So, after all the substitutions and simplifications, we find that the composite function (f ∘ g)(x) = 1 / (1 + 2x). It looks quite different from our original functions, doesn't it? This is the magic of composite functions – they create new relationships by combining existing ones.

Determining the Domain of (f ∘ g)(x)

Now that we've found the composite function, the next crucial step is to determine its domain. As we discussed earlier, the domain of (f ∘ g)(x) isn't just about the final function; it's also about the domains of the individual functions, f(x) and g(x), and how they interact.

Here's the breakdown:

  1. Find the domain of the inner function, g(x): g(x) = -1/x. The domain of g(x) is all real numbers except for x = 0, because division by zero is undefined. We can write this as: Domain(g) = {x | x ≠ 0}.
  2. Find the domain of the outer function, f(x): f(x) = x/(x-2). The domain of f(x) is all real numbers except for x = 2, because the denominator cannot be zero. We can write this as: Domain(f) = {x | x ≠ 2}.
  3. Consider the restrictions imposed by g(x) on f(x): This is the key step. We need to make sure that the output of g(x) does not fall outside the domain of f(x). In other words, we need to ensure that g(x) ≠ 2, because 2 is not in the domain of f(x).Let's solve the inequality g(x) ≠ 2: -1/x ≠ 2. Multiplying both sides by x (and being mindful of the sign if x were negative, but we'll address that with a separate case) gives us -1 ≠ 2x. Dividing by 2, we get x ≠ -1/2. So, we have another restriction: x cannot be -1/2.
  4. Find the domain of the composite function: Now, we combine all the restrictions we've found. The domain of (f ∘ g)(x) is all real numbers except for the values that violate any of our conditions: x ≠ 0 (from the domain of g), and x ≠ -1/2 (because g(x) cannot be 2).We can write the domain of (f ∘ g)(x) in set notation as:Domain(f ∘ g) = {x | x ≠ 0 and x ≠ -1/2}.

In interval notation, this would be: Domain(f ∘ g) = (-∞, -1/2) ∪ (-1/2, 0) ∪ (0, ∞).

Conclusion

Congratulations! You've successfully found the composite function (f ∘ g)(x) and determined its domain. We started with the functions f(x) = x/(x-2) and g(x) = -1/x, and after carefully substituting and simplifying, we found that (f ∘ g)(x) = 1 / (1 + 2x). Then, by considering the domains of both f(x) and g(x), we determined that the domain of (f ∘ g)(x) is all real numbers except for 0 and -1/2.

Remember, understanding composite functions and their domains is a crucial skill in mathematics. It allows you to model complex relationships and solve a wide range of problems. Keep practicing, and you'll become a master of composite functions in no time!

For more information on composite functions and related topics, you can visit Khan Academy's page on function composition.