Problem 84 - Intermediate Algebra

Question

If \(f(x)=x^{2}+1\) and \(g(x)=3 x^{2}-2,\) show that \((f \circ g)(x) \neq(g \circ f)(x)\).

Step-by-Step Solution

Answer

\((f \circ g)(x) \neq (g \circ f)(x)\) because they yield different polynomials.

Step 1: Understand the Composition

The composition \( (f \circ g)(x) \) means applying \( g(x) \) first and then applying \( f(x) \) to the result. Similarly, \( (g \circ f)(x) \) means applying \( f(x) \) first and then applying \( g(x) \) to the result.

Step 2: Find \((f \circ g)(x)\)

Start by finding \( f(g(x)) \). First, evaluate \( g(x) = 3x^2 - 2 \). Now plug this into \( f(x) \): \[ f(g(x)) = (3x^2 - 2)^2 + 1 \]Expand \((3x^2 - 2)^2 \): \[ (3x^2 - 2)^2 = (9x^4 - 12x^2 + 4) \]So, \[ f(g(x)) = 9x^4 - 12x^2 + 4 + 1 = 9x^4 - 12x^2 + 5 \]

Step 3: Find \((g \circ f)(x)\)

Now compute \( g(f(x)) \). Start with \( f(x) = x^2 + 1 \). Substitute this into \( g(x) \): \[ g(f(x)) = 3(x^2 + 1)^2 - 2 \]Expand \((x^2 + 1)^2 \): \[ (x^2 + 1)^2 = x^4 + 2x^2 + 1 \]Then, substitute into \( g(x) \): \[ g(f(x)) = 3(x^4 + 2x^2 + 1) - 2 = 3x^4 + 6x^2 + 3 - 2 \]Which simplifies to: \[ g(f(x)) = 3x^4 + 6x^2 + 1 \]

Step 4: Compare \((f \circ g)(x)\) and \((g \circ f)(x)\)

Now compare the two resulting expressions:\[ (f \circ g)(x) = 9x^4 - 12x^2 + 5 \]\[ (g \circ f)(x) = 3x^4 + 6x^2 + 1 \]Clearly, the coefficients of corresponding terms are different. Therefore, \((f \circ g)(x) eq (g \circ f)(x)\).

Key Concepts

Polynomial Functions

Polynomial functions form the backbone of algebra, providing a stable way to express a broad range of mathematical problems. They involve expressions with variables raised to whole number powers, and are very flexible for modeling real-world situations. When we look at the function like \( f(x) = x^2 + 1 \), we are exploring a common type called a quadratic polynomial. This type of function has major features like its parabolic shape and a vertex, which make it useful in various fields.

• Quadratic functions have the general form \( ax^2 + bx + c \), where \( a, b, c \) are constants.
• These functions graph into a symmetrical curve called a parabola.
• Polynomial terms are added, subtracted, and multiplied to form new expressions, providing a rich structure for solving equations.

In the exercise given, polynomial functions form the crux of the expressions \( f(x) = x^2 + 1 \) and \( g(x) = 3x^2 - 2 \). Expanding these functions using basic algebraic rules helps reveal deeper insights. Understanding how each term transforms and combines is crucial for efficiently solving more complex problems.

Function Notation

Function notation is a concise way of representing how variables are connected. It often simplifies the communication of mathematical computations and allows us to understand the order of operations involved. In notation like \( f(x) \) or \( g(x) \), the function name, such as \( f \) or \( g \), tells us what rule to apply to the variable \( x \). The outputs are determined based on these input values.

• It indicates the function and the variable being used, as in \( f(x) = x^2 + 1 \).
• The inside of the parentheses specifies the input, which can also be another function, as in function compositions.
• Changing the order of operations, especially in compositions like \( (f \circ g)(x) \) and \( (g \circ f)(x) \), impacts the final result.

Using function notation becomes especially powerful in function composition. This occurs when the result of one function becomes the input for another, allowing for the creation of more complex solutions. In this exercise, clear understanding of how to apply \( f \) and \( g \) using their definitions allows us to see why the compositions yield different results.

Mathematical Proof

Solid mathematical understanding often comes from carefully demonstrating why something is true through proofs. A common approach in solving problems like the one in this exercise is to provide a step-by-step proof, showcasing that two expressions are or aren't equal. With function compositions like \((f \circ g)(x)\) and \((g \circ f)(x)\), such proofs help clarify why these compositions of functions lead to different outcomes.

• The proof begins by explicitly calculating each composition, as shown in this exercise.
• Breaking down each step ensures that no part of the process is overlooked, which is crucial for clarity.
• The conclusion follows from a detailed comparison of the results, where differences in algebraic expressions become evident.

The importance of proving differences in math lies in verification and better comprehension of concepts. By methodically following through each series of operations, a proof not only answers the question but solidifies your grasp of the wider principles at play. As illustrated, showing that \((f \circ g)(x)\) is not equivalent to \((g \circ f)(x)\) deepens understanding and demonstrates the power of careful algebraic calculation.