Recall that a function f is odd if f(-x)=-f(x) or even if...

Recall that a function $f$ is odd if $f(-x)=-f(x)$ or even if $f(-x)=f(x)$ for all real $x$ (a) Show that a polynomial $P(x)$ that contains only odd powers of $x$ is an odd function. (b) Show that a polynomial $P(x)$ that contains only even powers of $x$ is an even function. (c) Show that if a polynomial $P(x)$ contains both odd and cven powers of $x,$ then it is neither an odd nor an even function. (d) Express the function $$P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5$$ as the sum of an odd function and an even function.

Recall that a function $f$ is odd if $f(-x)=-f(x)$ or even if $f(-x)=f(x)$ for all real $x$
(a) Show that a polynomial $P(x)$ that contains only odd powers of $x$ is an odd function.
(b) Show that a polynomial $P(x)$ that contains only even powers of $x$ is an even function.
(c) Show that if a polynomial $P(x)$ contains both odd and cven powers of $x,$ then it is neither an odd nor an even function.
(d) Express the function
$$P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5$$ as the sum of an odd function and an even function.

Key Concepts

Polynomial Terms with Specific Parity

A polynomial composed exclusively of odd powers is an odd function, as flipping the sign of the variable reverses the sign of each term. Conversely, a polynomial containing only even powers is an even function because the square, fourth power, etc., are invariant under the replacement of x by -x. When both even and odd powered terms are present, the function generally does not exhibit either perfect odd or even symmetry.

Odd Function

An odd function is defined by the property f(-x) = -f(x) for every x in its domain. This symmetry means that the graph of an odd function is rotationally symmetric about the origin. In polynomial terms, any function that is odd has only odd powers because the substitution x by -x will yield a sign change in each term, sustaining the odd function property.

Even Function

An even function satisfies the equality f(-x) = f(x) for all x in its domain, which implies that its graph is symmetric with respect to the y-axis. In the case of polynomials, if a function is even, all the terms involve even powers of x, since substituting -x for x does not affect the sign of these terms.

Decomposition into Odd and Even Components

Any function, including a polynomial, can be uniquely decomposed into the sum of an even function and an odd function. This decomposition is given by E(x) = [f(x) + f(-x)]/2 for the even part and O(x) = [f(x) - f(-x)]/2 for the odd part. This tool is fundamental in analysis and helps to isolate symmetrical properties of functions.

Transcript

00:02 All right, we want to show that a polynomial with only odd powers must be odd.

00:11 And first, let's talk about the definition of an odd function.

00:17 If you evaluate a function at negative x and it equals negative f of x, then we know the function is odd.

00:28 So if we were to create some function right now, let's say f of x is equal to 2.

00:34 2x to the 5th power minus 3x cubed plus x.

00:43 And i evaluate this function for negative x.

00:51 So if i evaluate f of negative x, for example, then i will have 2 times negative x to the 5th power minus 3 times negative x cubed plus negative x when i make all my substitutions, then this will simplify, right, negative x to the fifth power will be negative x to the fifth times two.

01:14 That'll be negative 2x to the fifth.

01:18 This will be negative 3 times negative x cubed.

01:21 That's going to be plus 3x cubed minus x.

01:26 That is f of negative x.

01:29 And if we factor a negative 1 out right now, then i get negative the quantity 2x to the 5th minus 3x cubed plus x, which you'll notice is f of x.

01:46 So we have now shown that f of negative x equals negative f of x, and therefore this function that has only powers of x that are odd is an odd function.

02:00 It fits the definition of an odd function.

02:06 The next thing we want to show is that a polynomial with only even powers must be an even function.

02:17 So the definition of an even function, if we evaluate a function at negative x, and we get the original function back as a result, then the function must be even.

02:30 So an example of a function that has only even exponents in it, if i did something like x to the fourth minus 2x squared plus 7, let's say, 7 technically is 7x to 3 .5 .000 technically is 7x to 3.

02:51 The 0 power, so again that's a power of x that's even.

02:55 When i evaluate f of negative x, i will get negative x to the fourth power minus 2 times negative x squared plus 7...

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