Math Revision 16.7 - Even and Odd Functions

In these revision notes for Even and Odd Functions, we cover the following key points:

• What are even functions?
• What is the condition for a function to be even?
• How to check whether a function is even or not by substitution?
• How to understand whether a function is even or not without substitutions?
• What are the features of the graph of an even function?
• What are the features of the graph of an odd function?
• What happens if a function is neither even nor odd?
• What are the properties of even functions?
• What are the properties of odd functions?

Even and Odd Functions Revision Notes

By definition, even functions are those functions for which f(x) = f(-x) for every x.

We cannot rely on numerical substitutions to confirm whether a function is even or not, but it is necessary to use a general method for this. Indeed, it is clear that if we are able to obtain two identical expressions for f(-x) and f(x), then the function f(x) is even. In this way, no numerical substitutions are necessary to prove the evenness of a function. This reasoning brings the general rule of even functions:

If f(x) = f(-x) for any x-value, then the function f(x) is even. In other words, a function is even if and only if f(x) - f(-x) = 0 for any x.

Since in even functions f(x) is always equal to f(-x), the left part of the graph is symmetrical to the right part, where the Y-axis acts as a symmetry axis. We have called it horizontal symmetry. Therefore, we can understand whether a given function is even or not by checking the horizontal symmetry of the graph.

By definition, an odd function f(x) is a function for which f(-x) is always equal to -f(x). In other words, If f(-x) - [- f(x)] = 0 for any x-value, then the function f(x) is odd.

Substituting certain x-values in a function's formula does not give us a definitive answer regarding the oddness of the given function. The best method for this is to check whether f(-x) = -f(x) for any x. In other words, we must check whether f(-x) - [-f(x)] = 0. If yes, the function is odd; otherwise, it is not odd.

Since in odd functions f(-x) = -f(x) for any x, if for example f(x) is located at the upper-right side of the coordinates system (first quadrant), then f(-x) lies on the upper left side of the coordinates system (second quadrant). On the other hand, -f(x) lies on the bottom-right side (fourth quadrant), so f(-x) and -f(x) lie on the opposite parts of the coordinates system. Making half a rotation of f(-x) around the origin gives -f(x). This feature distinguishes odd functions from non-odd ones.

We can draw the following conclusions about the evenness/oddness of a function f(x):

• Even functions must be sought among even-degree polynomials while odd functions among odd-degree polynomials. For example, some linear functions (first-degree polynomial functions) can be odd but none of the second-degree (quadrati
• polynomial functions can be odd.
• Linear functions of the general form f(x) = ax + b are odd only for b = 0, i.e. when the graph passes through the origin.
• Quadratic functions of the general form f(x) = ax² + bx + c are even only for b = 0, i.e. when the vertex lies on the vertical axis.
• Cubic functions of the general form f(x) = ax³ + bx² + cx + d are odd only for b = 0 and d = 0, i.e. when the graph passes through the origin.
• Biquadratic functions (fourth-degree polynomial functions) of the general form f(x) = ax⁴ + bx³ + cx² + dx + e is even only for b = 0 and d = 0, i.e. when the middle local minimum/maximum passes through the vertical axis.
• In general, odd-degree polynomial functions are only odd if the coefficients of the even-degree terms are 0 (i.e. if the even-degree terms do not exist in that function), while even-degree polynomial functions are even only if the coefficients of the odd-degree terms are 0 (i.e. if the odd-degree terms do not exist in that function).

Most functions are neither even nor odd. If the question requires studying a function, one of the elements to study involves checking the evenness and the oddness of that function.

If two even functions f(x) and g(x) are given, the following properties are true:

1. The sum of two even functions is always an even function. In symbols,
   f(x) = even & g(x) = even ⟹ f(x) + g(x) = even
2. The difference of two even functions is always an even function. In symbols,
   f(x) = even & g(x) = even ⟹ f(x) - g(x) = even
3. The product of two even functions is always an even function. In symbols,
   f(x) = even & g(x) = even ⟹f(x) ∙ g(x) = even
4. The quotient of division of two even functions is always an even function. In symbols,
   f(x) = even & g(x) = even ⟹ f(x) ÷ g(x) = even
5. The composition of two even functions is always an even function. In symbols,
   f(x) = even & g(x) = even ⟹ f∘g(x) = even and g∘f(x) = even

The properties of odd functions are similar to those of even functions. Thus, if two odd functions f(x) and g(x) are given, the following properties are true:

1. The sum of two odd functions is always an odd function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x) + g(x) = odd
2. The difference of two odd functions is always an odd function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x)-g(x) = odd
3. The product of two odd functions is always an even function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x) ∙ g(x) = even
4. The quotient of division of two odd functions is always an even function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f(x) ÷ g(x) = even
5. The composition of two odd functions is always an odd function. In symbols,
   f(x) = odd & g(x) = odd ⟹ f∘g(x) = odd and g∘f(x) = odd
6. The composition of an even and an odd function is always even. In symbols,
   f(x) = odd & g(x) = even ⟹ f∘g(x) = even and g∘f(x) = even
   or
   f(x) = even & g(x) = odd ⟹f∘g(x) = even and g∘f(x) = even