Problem 112 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove that a function of the following form is even. $$y=a_{2 n} x^{2 n}+a_{2 n-2} x^{2 n-2}+\cdots+a_{2} x^{2}+a_{0}$$

Step-by-Step Solution

Answer

The function is even since after substiting -x for x and simplifying, the function remains the same as the original function. This is in accordance with the definition of even functions, hence proven.

Step 1: Substitute -x for x

Replace each occurrence of x in the original function with -x.\n\[y = a_{2 r}(-x)^{2 n}+a_{2 n-2}(-x)^{2 n-2}+...+a_{2}(-x)^{2}+a_{0}\]

Step 2: Simplify the equation

In this step, we understand that (-x)^2n equals to x^(2n) as when we raise -x to an even power, the result is always positive which is same as x raised to 2n.\n\[y = a_{2 r}(x)^{2 n} + a_{2 n-2}(x)^{2 n-2} + ... + a_{2}(x)^{2} + a_{0}\]

Step 3: Compare the result with the original function

We see here that the given function and the simplified function after replacing x with -x are the same. As a result, we can conclude that the function is even.

Key Concepts

Even Function Properties

When we talk about even functions, we're discussing a very specific kind of symmetry in mathematics. Specifically, a function is even if for every number 'x' in the function's domain, the value of the function at 'x' is the same as its value at '-x'. Mathematically, we describe this property as:
\[f(x) = f(-x)\]
This definition means that if you were to graph an even function, you'd find that its shape is symmetrical across the y-axis. This is because the points (x, f(x)) and (-x, f(x)) have the same value for 'y' and are thus reflections of each other across the vertical axis.

• An example of an even function is the quadratic function \(f(x) = x^2\).
• To determine if a function is even, perform a simple test by substituting '-x' for 'x' and simplifying. If you get back the original function, it's even!
• Standard polynomial terms like \(x^2, x^4, x^6\), and so on, with even exponents are individually even functions, which leads to a sum of such terms being even as well.

In our exercise, we examined a polynomial function, which exhibited evenness due to all its terms being powers of even exponents. By proving that the function remains unchanged when \(x\) is replaced by \(-x\), we've confirmed that our function is indeed even, showcasing the distinct property of even functions.

Polynomial Functions

Diving into the realm of polynomial functions, we find that they are algebraic expressions involving a sum of powers in one variable. The general form of a polynomial function in 'x' is:
\[P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\]
where each \(a_i\) represents a coefficient and 'n' is a non-negative integer representing the degree of the polynomial, which is the highest power of 'x' in the expression.

Characteristics of Polynomial Functions:

• Polynomials are continuous and smooth; there are no sharp corners or breaks in their graphs.
• The degree of the polynomial determines the number of roots, or solutions, the equation has.
• The leading term \(a_nx^n\) will often dictate the end behavior of the polynomial's graph.

In our proof exercise, we've dealt with a polynomial function composed solely of even-power terms, which naturally leads to an even function, provided all coefficients are real numbers. The polynomial's evenness indicates that its graph is symmetric relative to the y-axis, which is a useful visualization for understanding these functions.

Power of Negative Numbers

Understanding the power of negative numbers is crucial when analyzing even functions, as it's often involved in proving a function's symmetry. The key principle here is that when a negative number is raised to an even power, the result is positive. Conversely, raising a negative number to an odd power yields a negative result. This idea is critical in simplifying functions during the process of proving evenness.
For example:

• \((-3)^2 = 9\), a positive result because 2 is even,
• While \((-3)^3 = -27\), a negative result because 3 is odd.

This property comes into play in Step 2 of our textbook solution, where we simplify \((-x)^{2n}\) to \(x^{2n}\). Since \(2n\) is an even integer—regardless of what \(n\) is—the negative sign is effectively 'canceled out,' confirming that an even powered polynomial component always contributes to the evenness of the function. This principle underscores the importance of recognizing the powers involved in a function to ascertain its symmetry and behavior.