Identifying Even, Odd, and Neither Functions: A Guide for Students

This article aims to provide a comprehensive understanding of how to determine whether a function is even, odd, or neither, using examples and explanations suitable for a grade 10 student and beyond.

Introduction to Even and Odd Functions

When we talk about functions, an even function is one where replacing x with -x results in the same function. In mathematical terms, this is expressed as:

Even Function Definition

f(x) f(-x)

An odd function is one where replacing x with -x results in the negative of the function. This is expressed mathematically as:

Odd Function Definition

f(-x) -f(x)

Examples and Practice

Example 1: Determining if a Function is Even, Odd, or Neither

Consider the function p(x) 12^2 3^2.

Working for p(x)

p(1) 12^2 3^2 144 9 153 p(-1) -12^2 1^2 -144 1 -143

The results are not the same, so p(x) is not even. Since these results are not the same and not opposite numbers, p(x) is not odd. Therefore, p(x) is neither even nor odd.

Example 2: Determining if a Function is Even, Odd, or Neither (Another Method)

Consider the function q(x) 25 / (3 - 1).

Working for q(x) q(1) 25 / (3 - 1) 25 / 2 12.5 q(-1) -25 / (-3 - 1) -25 / -4 6.25

Since these results are not the same and not opposite numbers, q(x) is neither even nor odd.

Categorization of Functions

Some well-known even functions include:

Even Functions

f(x) x^2 f(x) cos(x)

Some well-known odd functions include:

Odd Functions

f(x) x^3 f(x) sin(x) f(x) tan(x)

Further Examples and Reasoning

Example 3: Function Composition

Consider the function a(x) x^5 and the function x (which is x 1) to see how the function behaves with respect to evenness and oddness:

Function a(x) is odd x (1) is also odd

Since both components are odd, a(x) x^5 is odd.

Example 4: Polynomial Functions

Consider the function g(x) 1 - x^4:

Working for g(x)

g(1) 1 - 1^4 1 - 1 0 g(-1) 1 - (-1)^4 1 - 1 0

Since g(-x) g(x), g(x) 1 - x^4 is an even function.

Example 5: Mixed Example

Consider the function h(x) 2x - x^2:

Working for h(x)

h(1) 2(1) - (1)^2 2 - 1 1 h(-1) 2(-1) - (-1)^2 -2 - 1 -3

Since h(x) 2x - x^2 and -h(x) -2x x^2 do not result in equivalent expressions, h(x) is neither even nor odd.

Conclusion

This article has provided a clear understanding of even, odd, and neither functions with practical examples and reasoning. Using these definitions and rules, students can effectively determine the nature of a given function. By practicing with different types of functions, the student can build a strong foundation in recognizing and working with even, odd, and neither functions.