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Chapter 0: Problem 98

$\begin{array}{l}{\text { Proof Prove that the function is even. $$f(x)=a_{2 \pi} x^{2 n}+a_{2 n-2} x^{2 n-2}+\cdots+a_{2} x^{2}+a_{0}$$

Short Answer

Expert verified
The function \( f(x) = a_{2n} x^{2n} + a_{2n-2} x^{2n-2} + \ldots + a_{2} x^{2} + a_{0} \) is an even function because it meets the property \( f(x) = f(-x) \) for all \( x \).

Step by step solution

01

Understand an Even Function

A function is odd or even if it satisfies certain symmetry properties. In particular, a function is even if its graph is symmetric with respect to the y-axis. Mathematically, a function \( f(x) \) is even if, for any \( x \) in the function's domain, \( f(x) = f(-x) \). This is the property we will use to prove that the given function is even.
02

Substitute -x into the Function

First, write down the function \( f(x) \) and then replace each \( x \) with \( -x \). We'll get: \( f(-x) = a_{2n}(-x)^{2n} + a_{2n-2}(-x)^{2n-2} + \ldots + a_{2}(-x)^{2} + a_{0} \).
03

Simplify the Result

Next, simplify the result. Every term in the function has exponent as a multiple of 2. Since the negative sign gets eliminated when raised to even powers, the function simplifies back to the original function: \( f(-x) = a_{2n} x^{2n} + a_{2n-2} x^{2n-2} + \ldots + a_{2} x^{2} + a_{0} = f(x) \).
04

Conclude the Proof

In step 3, we simplified \( f(-x) \) to be identical to \( f(x) \). That means the condition \( f(x) = f(-x) \) for all \( x \) is fulfilled. Therefore, the function \( f(x) = a_{2n} x^{2n} + a_{2n-2} x^{2n-2} + \ldots + a_{2} x^{2} + a_{0} \) is an even function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Even Functions
The concept of an even function is deeply linked to the idea of symmetry in mathematics. Specifically, an even function showcases a mirror-like symmetry across the y-axis. To formally identify an even function, we examine the equation of the function when we substitute every instance of the variable x with -x. If after simplifying the equation, it remains unchanged, that is, if
\( f(x) = f(-x) \)
for every x value in the domain of the function, we can conclude that the function is even.

This property can be visually appreciated by looking at the graph of the function. Common examples of even functions include x raised to any even power, such as x^2 or x^4, and the cosine function. This symmetry principle simplifies many aspects of mathematical analysis and problem-solving, allowing us to predict the behavior of functions and their graphs.
Exploring Polynomial Functions
A polynomial function is a mathematical expression consisting of variables and coefficients, structured in a specific manner where the variables are raised to only non-negative integer exponents. Polynomial functions take the form
\( p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \)
where \( a_n, a_{n-1}, ..., a_0 \) are constants with \( a_n \) not equal to zero and n is a non-negative integer which represents the degree of the polynomial.

Polynomials are foundational in algebra and appear in many different aspects of mathematics and applied science. Their properties, such as end-behavior, turning points, and the location of their zeros, play a crucial role in understanding the overall shape and behavior of their graphs. Since polynomial terms with even exponents contribute to the even nature of a function, as seen in the original exercise, recognizing the structure of polynomials is essential for identifying symmetry properties.
The Role of Mathematical Proof
A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. It involves a sequence of logical deductions starting from known facts and axioms. Mathematical proofs are essential in verifying theorems, propositions, and mathematical concepts, lending credibility and certainty to mathematical discourse.

In the context of the given exercise, a proof was constructed to demonstrate that a particular polynomial function is even. This proof followed a structured approach, beginning with the definition of an even function, followed by the substitution and simplification steps, and ending with the conclusion that confirms the initial statement. Proofs can be direct, as in this case, or indirect and may involve various techniques, such as induction, contradiction, or contrapositive arguments. Understanding how to construct and interpret proofs is a cornerstone of advanced mathematical study and practice.

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