# Is this an even or odd function

1. (sin4x + cos4x)

## The Attempt at a Solution

(sin4x + cos4x)
= (sin4(-x) + cos4(-x))
= -sin4x + cos4x

im thinking it is an even function

## Answers and Replies

For an even function, f(-x) = f(x). An odd function has f(-x) = -f(x). Is your final result equal to either f(x) or -f(x) ? Note that most functions are neither even nor odd.

Mark44
Mentor
1. (sin4x + cos4x)

## The Attempt at a Solution

(sin4x + cos4x)
= (sin4(-x) + cos4(-x))
= -sin4x + cos4x
You started off by saying sin(4x) + cos(4x) = sin(-4x) + cos(-4x), which your later work shows isn't true. Your work should start with sin(-4x) + cos(-4x). If you can show this is equal to sin(4x) + cos(4x), then the function is even. If it turns out to be equal to -sin(4x) - cos(4x), then the function is odd. If it results in neither, then the function is neither even nor odd.
im thinking it is an even function

so is it an odd function since it does not look like the original function?

If you want to decide whether a function f is odd, you can start by looking at f(0). If that's ≠0, the function definitely isn't odd. (It could still be even though.)

so is it an odd function since it does not look like the original function?

A function is odd if it has a very specific behavior. For every value of x, f(-x) must equal -f(x). Find the expression for f(-x) and find the expression for -f(x). Compare the two expressions. Are they equal for every possible value of x? If they are, your function is odd. If they differ for any single value of x, your function is not odd.
A quick way to compare two expressions is to subtract one from the other. It doesn't matter which one comes first, as if they are the same value, the subtraction will give you 0. If you don't get 0, the expressions are different.
If your function is not odd, that does not mean it is even. To be even it must satisfy a different extremely specific rule. You must compare the expression for f(-x) to the expression for f(x). Are they equal for every possible value of x? If so, the function is even. If they differ for any specific value of x, then your function is not even either.
Most functions are neither even nor odd.

Last edited:
HallsofIvy
Science Advisor
Homework Helper
so is it an odd function since it does not look like the original function?
You appear to thinking that any function that is not "even" must be "odd". That is not true. slider142, in the first response to your post, told you that a function is odd if and only if f(-x)= -f(x). slider142 also told you that "most functions are neither even nor odd."

When the concepts of "even functions" and "odd functions" are taught, it might be best to tell students that most functions are neither. Just like most numbers are neither even nor odd, for instance 1.5 isn't, nor is pi.