Even and odd functions have significance in graphical analysis, especially of trigonometric functions. The concept is relatively simple, and based on certain symmetries of the graphs. An even function will have perfect symmetry relative to the y-axis. At the same time, an odd function will have perfect symmetry relative to the origin.
Here is a basic example to illustrate the properties of even functions.
Example 1:
If \(n\) is an even number, show that \(x^{2n}\) is an even function.
Solution:
By exponent rules:
$$x^{2n} = (x^2)^n$$
Even functions are, by definition, symmetric relative to the y-axis, so
$$f(x) = f(-x)$$
Our function is \(f(x) = x^{2n}\). To show that it is even, compare \(f(x)\) to \(f(-x)\):
$$(x^n)^2 = ((-x)^n)^2 \Rightarrow$$ $$(x^2)^n = ((-x)^2)^n \Rightarrow$$ $$x^{2n} = x^{2n}$$
Another thing to realize is that the trigonometric functions are some of the most common even and odd functions we work with in mathematics.
The cosine function is an even function; it is symmetric relative to the y-axis.
Here is an example to illustrate how odd functions work.
Example 2: Show that \(f(x) = x^7\) is an odd function.
Solution: For \(f(x)\) to be an odd function, it must be symmetric relative to the origin, so
$$f(-x) = -f(x)$$
Plug this into the given function:
$$(-x)^7 = -x^7$$
Rewrite the left side:
$$(-1 \cdot x)^7 = -x^7 \Rightarrow$$ $$-1^7 \cdot x^7 = -x^7 \Rightarrow$$ $$-1 \cdot x^7 = -x^7 \Rightarrow$$ $$-x^7 = -x^7$$
Cosine is an even function. Interestingly enough:
The sine function is an odd function, because it is symmetric relative to the origin.
In many problems it will be necessary to work with both even and odd functions. Here are some examples.
Example 3: What is the one function (in the Cartesian Coordinate System) that is both even and odd?
Solution: Since even functions are defined as
$$f(-x) = f(x)$$
and odd functions are defined as
$$f(-x) = -f(x),$$
by the Transitive Property of Equality, we can substitute to get
$$f(x) = -f(x)$$
The only function that satisfies this equation is $$f(x) = 0.$$ Graphing the function will verify that it is both even and odd.
Example 4: Based on a graph of \(f(x) = \sqrt{x}\), why can it not be even or odd?
Solution: \(f(x)\) is only defined for nonnegative numbers, which means on the graph of \(f(x)\) on the Cartesian Plane, all the space to the left of the y-axis is completely empty. Therefore the graph cannot be symmetric relative to the y-axis or the origin, and it is not even or odd.
The next pair of examples shows a pair of interesting properties of even and odd functions.
Example 5: Show that the product of two odd functions is an even function.
Solution: Let \(h(x)\) be defined so that
$$h(x) = f(x) \cdot g(x)$$
The statement of the problem then implies the following three statements based on the definitions of even and odd functions:
$$h(-x) = h(x)$$ $$f(-x) = -f(x)$$ $$g(-x) = -g(x)$$
We want to prove that the product of the two odd functions is equal to the even function, so set up an equation representing this:
$$h(-x) = f(-x) \cdot g(-x)$$
\(h(x)\) is an even function, so this equation becomes
$$h(x) = f(-x) \cdot g(-x)$$
Since \(f(x)\) and \(g(x)\) are odd functions, a similar substitution can be made on the right-hand side of the equation:
$$h(x) = -f(x) \cdot -g(x) \Rightarrow$$ $$h(x) = f(x) \cdot g(x)$$
This is the original statement we used to define the three functions in terms of each other; therefore our theorem is true.
Example 6: Prove that the product of an even function and another even function is an even function.
Solution: The process for this solution is very similar to the previous one. Let the functions \(f(x)\), \(g(x)\), and \(h(x)\) be defined so that
$$h(x) = f(x) \cdot g(x)$$
If all three functions are even, that means the following equation, derived from the original equation, has a special property:
$$h(-x) = f(-x) \cdot g(-x)$$
Since all three functions are even, each argument can change its sign freely. In particular we get
$$h(x) = f(x) \cdot g(x)$$
This is what we defined at the beginning, so it is true that the product of two even functions is an odd function.