Function Composition And Polynomial Equivalency F(x) = X + 1 And G(x) = 2/x

In the realm of mathematics, the composition of functions stands as a pivotal concept, allowing us to combine multiple functions to create new ones. This operation, often denoted as (f ∘ g)(x), essentially means applying function g to x first, and then applying function f to the result. In simpler terms, we're feeding the output of one function as the input to another. This powerful technique finds widespread applications in various fields, including calculus, analysis, and computer science.

Understanding the Mechanics of Function Composition

At its core, function composition involves a sequential application of functions. To illustrate, consider two functions, f(x) and g(x). The composition (f ∘ g)(x) is evaluated by first determining g(x), and then substituting this value into f(x). Mathematically, this can be expressed as (f ∘ g)(x) = f(g(x)). This notation highlights the order of operations: g is applied first, followed by f. Let's delve into a concrete example to solidify this concept. Suppose f(x) = x + 1 and g(x) = x². To find (f ∘ g)(x), we first evaluate g(x), which yields x². We then substitute x² into f(x), resulting in f(x²) = x² + 1. Thus, (f ∘ g)(x) = x² + 1. The order of composition matters significantly. The composition (g ∘ f)(x) is generally different from (f ∘ g)(x). In our example, (g ∘ f)(x) would be g(f(x)) = g(x + 1) = (x + 1)², which is distinct from x² + 1. This non-commutativity underscores the importance of carefully considering the order of function application.

Polynomials: The Building Blocks of Algebraic Expressions

Polynomials are fundamental algebraic expressions that play a crucial role in mathematics. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. A general form of a polynomial in a single variable, x, can be written as: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x¹ + a₀, where aₙ, aₙ₋₁, ..., a₁, a₀ are constants (coefficients) and n is a non-negative integer (the degree of the polynomial). Polynomials exhibit a range of desirable properties that make them amenable to mathematical manipulation. They are continuous, smooth, and have well-defined derivatives and integrals. These properties make them invaluable tools in modeling real-world phenomena and solving mathematical problems. Polynomials can be classified based on their degree, which is the highest power of the variable. For instance, a linear polynomial has a degree of 1, a quadratic polynomial has a degree of 2, and a cubic polynomial has a degree of 3. The coefficients of a polynomial determine its specific characteristics. The leading coefficient (aₙ) is particularly important, as it influences the polynomial's end behavior. The constant term (a₀) represents the y-intercept of the polynomial's graph.

The Interplay of Function Composition and Polynomials

When we compose functions, it's natural to ask whether the resulting function is also a polynomial. This depends on the nature of the functions being composed. If both functions are polynomials, their composition will also be a polynomial. However, if one or both functions are not polynomials, the composition may or may not be a polynomial. For example, composing two linear functions will always result in a linear function, which is a polynomial. However, composing a polynomial with a rational function (a function that is a ratio of two polynomials) may not yield a polynomial. Consider the functions f(x) = x² + 1 (a polynomial) and g(x) = 1/x (a rational function). The composition (f ∘ g)(x) = f(g(x)) = f(1/x) = (1/x)² + 1 = 1/x² + 1, which is not a polynomial due to the negative exponent of x. In summary, the composition of functions can lead to a variety of outcomes, and determining whether the result is a polynomial requires careful analysis of the functions involved.

In this problem, we are presented with two functions: f(x) = x + 1 and g(x) = 2/x. Our objective is to determine the polynomial that is equivalent to the composition (f ∘ g)(x). To achieve this, we will meticulously follow the process of function composition, substituting g(x) into f(x) and simplifying the resulting expression. Let's embark on this journey step by step. First, we need to find (f ∘ g)(x), which means we need to evaluate f(g(x)). We know that g(x) = 2/x, so we substitute this into f(x): f(g(x)) = f(2/x). Now, we replace x in f(x) with 2/x: f(2/x) = (2/x) + 1. To simplify this expression, we need to combine the terms. We can rewrite 1 as x/x to have a common denominator: (2/x) + 1 = (2/x) + (x/x). Now, we can add the fractions: (2/x) + (x/x) = (2 + x) / x. This gives us the composition (f ∘ g)(x) = (2 + x) / x or (x+2)/x. This resulting expression is a rational function, which is a ratio of two polynomials. The numerator is the polynomial x + 2, and the denominator is the polynomial x. Thus, we have found the equivalent expression for (f ∘ g)(x).

Identifying the Equivalent Polynomial

Our next step is to identify which of the provided options matches the result we obtained, which is (x + 2)/x. Let's examine the given options:

A. 2/(x + 1) B. (2x + 2)/x C. Comparing Our Result with the Options

By carefully comparing the result we derived, (x + 2)/x, with the provided options, we can pinpoint the equivalent polynomial. Option A, 2/(x + 1), is clearly different from our result. Option B, (2x + 2)/x, can be simplified by factoring out a 2 from the numerator: (2x + 2)/x = 2(x + 1)/x. This is also different from our result, (x + 2)/x. Option C is missing from the provided options. However, based on our derivation, the correct expression for (f ∘ g)(x) is (x + 2)/x.

In this exploration, we have delved into the concept of function composition, a fundamental operation in mathematics. We've seen how composing functions involves applying one function to the result of another, creating a new function. We specifically examined the composition (f ∘ g)(x) where f(x) = x + 1 and g(x) = 2/x, meticulously deriving the equivalent expression (x + 2)/x. This process highlights the importance of order in function composition and the techniques for simplifying resulting expressions. Furthermore, we reinforced our understanding of polynomials and their role in algebraic expressions. By comparing our derived result with the provided options, we demonstrated the ability to identify equivalent expressions, a crucial skill in mathematical problem-solving. This exercise not only solidifies our grasp of function composition and polynomial equivalence but also underscores the interconnectedness of mathematical concepts. As we continue our mathematical journey, these foundational principles will serve as invaluable tools in tackling more complex challenges.